View source for Algebra

{{Short description|Branch of mathematics}}
{{About||the kind of algebraic structure|Algebra over a field|other uses}}
{{Featured article}}
{{Use mdy dates|date=August 2024}}
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|image1=Polynomial2.svg
|alt1=Polynomial equation
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|caption1=[[Elementary algebra]] studies which values solve equations formed using arithmetical operations.
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|alt2=Signature of the ring of integers
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|caption2=[[Abstract algebra]] studies [[algebraic structure]]s, such as the [[ring of integers]] given by the set of [[integer]]s <math>(\Z)</math> together with [[Algebraic operation|operations]] of [[addition]] (<math>+</math>) and [[multiplication]] (<math>\times</math>).
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'''Algebra''' is the branch of [[mathematics]] that studies certain abstract [[system]]s, known as [[algebraic structures]], and the manipulation of [[Expression (mathematics)|expressions]] within those systems. It is a generalization of [[arithmetic]] that introduces [[Variable (mathematics)|variables]] and [[algebraic operation]]s other than the standard arithmetic operations, such as [[addition]] and [[multiplication]].

[[Elementary algebra]] is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. [[Linear algebra]] is a closely related field that investigates [[linear equation]]s and combinations of them called ''[[system of linear equations|systems of linear equations]]''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions.

[[Abstract algebra]] studies algebraic structures, which consist of a [[Set (mathematics)|set]] of [[mathematical objects]] together with one or several [[Operation (mathematics)|operations]] defined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]], based on the number of operations they use and the laws they follow, called [[axioms]]. [[Universal algebra]] and [[category theory]] provide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures.

Algebraic methods were first studied in the [[ancient period]] to solve specific problems in fields like [[geometry]]. Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until the 16th and 17th centuries when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond a [[theory of equations]] to cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry, [[topology]], [[number theory]], and [[calculus]], and other fields of inquiry, like [[logic]] and the [[empirical sciences]].

== Definition and etymology ==
Algebra is the branch of mathematics that studies [[algebraic structure]]s and the [[algebraic operations|operations]] they use.<ref>{{multiref | {{harvnb|Merzlyakov|Shirshov|2020|loc=Lead section}} | {{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 4]}} }}</ref> An algebraic structure is a non-empty [[Set (mathematics)|set]] of [[mathematical object]]s, such as the [[integer]]s, together with algebraic operations defined on that set, like [[addition]] and [[multiplication]].<ref>{{multiref | {{harvnb|Fiche|Hebuterne|2013|p=[https://books.google.com/books?id=TqkckiuuXg8C&pg=PT326 326]}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.}} | {{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 4]}} }}</ref>{{efn|When understood in the widest sense, an algebraic operation is a [[function (mathematics)|function]] from a [[Cartesian product#n-ary Cartesian power|Cartesian power of a set into that set]], expressed formally as {{nowrap|<math>\omega: A^n \to A</math>.}} The addition of real numbers is an example of an algebraic operation: it takes two numbers as input and produces one number as output. It has the form {{nowrap|<math>+: \R^2 \to \R</math>.<ref>{{harvnb|Baranovich|2023|loc=Lead section}}</ref>}}}} Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines the use of [[Variable (mathematics)|variables]] in [[Algebraic equation|equations]] and how to manipulate these equations.<ref>{{multiref | {{harvnb|Pratt|2022|loc=Lead section, § 1. Elementary Algebra, § 2. Abstract Algebra, § 3. Universal Algebra}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.}} }}</ref>{{efn|Algebra is covered by division 512 in the [[Dewey Decimal Classification]]<ref>{{harvnb|Higham|2019|p=[https://books.google.com/books?id=ferEDwAAQBAJ&pg=PA296 296]}}</ref> and subclass QA 150{{hyphen}}272.5 in the [[Library of Congress Classification]].<ref>{{harvnb|Library of Congress|p=3}}</ref> It encompasses several areas in the [[Mathematics Subject Classification]].<ref>{{harvnb|zbMATH Open|2024}}</ref>}}

Algebra is often understood as a generalization of [[arithmetic]].<ref>{{multiref | {{harvnb|Maddocks|2008|p=129}} | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA45 45]}} }}</ref> Arithmetic studies operations like addition, [[subtraction]], multiplication, and [[Division (mathematics)|division]], in a particular domain of numbers, such as the real numbers.<ref name="auto1">{{multiref | {{harvnb|Romanowski|2008|pp=302–303}} | {{harvnb|HC Staff|2022}} | {{harvnb|MW Staff|2023}} | {{harvnb|Bukhshtab|Pechaev|2020}} }}</ref> [[Elementary algebra]] constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in the form of variables in addition to numbers.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=129–130}} | {{harvnb|Pratt|2022|loc=Lead section, § 1. Elementary Algebra}} | {{harvnb|Wagner|Kieran|2018|p=[https://books.google.com/books?id=uW4ECwAAQBAJ&pg=PT225 225]}} }}</ref> A higher level of abstraction is found in [[abstract algebra]], which is not limited to a particular domain and examines algebraic structures such as [[Group (mathematics)|groups]] and [[Ring (mathematics)|rings]]. It extends beyond typical arithmetic operations by also covering other types of operations.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead section, § 2. Abstract Algebra}} | {{harvnb|Wagner|Kieran|2018|p=[https://books.google.com/books?id=uW4ECwAAQBAJ&pg=PT225 225]}} }}</ref> Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates the characteristics of algebraic structures in general.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3. Universal Algebra}} | {{harvnb|Grillet|2007|p=[https://link.springer.com/chapter/10.1007/978-0-387-71568-1_15 559]}} | {{harvnb|Denecke|Wismath|2018|p=[https://books.google.com/books?id=2UlZDwAAQBAJ&pg=PR5 v]}} | {{harvnb|Cohn|2012|p=[https://books.google.com/books?id=6tbuCAAAQBAJ&pg=PR13 xiii]}} }}</ref>

[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|upright=.8|thumb|alt=Title page of The Compendious Book on Calculation by Completion and Balancing|The word ''algebra'' comes from the title of [[al-Khwarizmi]]'s book {{transliteration|ar|[[Al-Jabr]]}}.<ref>{{multiref | {{harvnb|Cresswell|2010|p=[https://books.google.com/books?id=J4i3zV4vnBAC&pg=PA11 11]}} | {{harvnb|OUP Staff}} | {{harvnb|Menini|Oystaeyen|2017|p=722}} }}</ref>]]

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.<ref>{{multiref | {{harvnb|Weisstein|2003|p=46}} | {{harvnb|Walz|2016|loc=[https://www.spektrum.de/lexikon/mathematik/algebra/12062 Algebra]}} }}</ref> When used as a [[countable noun]], an algebra is [[algebra over a field|a specific type of algebraic structure]] that involves a [[vector space]] equipped with [[Bilinear map|a certain type of binary operation]].<ref>{{multiref | {{harvnb|Weisstein|2003|p=46}} | {{harvnb|Brešar|2014|p=[https://books.google.com/books?id=PkvPBAAAQBAJ&pg=PR33 xxxiii]}} | {{harvnb|Golan|1995|pp=[https://link.springer.com/chapter/10.1007/978-94-015-8502-6_18 219–227]}} }}</ref> Depending on the context, "algebra" can also refer to other algebraic structures, like a [[Lie algebra]] or an [[associative algebra]].<ref>{{harvnb|EoM Staff|2017}}</ref>

The word ''algebra'' comes from the [[Arabic]] term {{lang|ar|الجبر}} ({{transliteration|ar|al-jabr}}), which originally referred to the surgical treatment of [[bonesetting]]. In the 9th century, the term received a mathematical meaning when the Persian mathematician [[Muhammad ibn Musa al-Khwarizmi]] employed it to describe a method of solving equations and used it in the title of a treatise on algebra, {{transliteration|ar|[[Al-Jabr|al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah]]}} [''The Compendious Book on Calculation by Completion and Balancing''] which was translated into Latin as {{lang|la|Liber Algebrae et Almucabola}}.{{efn|The exact meaning of the term {{transliteration|ar|al-jabr}} in al-Khwarizmi's work is disputed. In some passages, it expresses that a quantity diminished by subtraction is restored to its original value, similar to how a bonesetter restores broken bones by bringing them into proper alignment.<ref>{{multiref | {{harvnb|Oaks|Alkhateeb|2007|pp=45–46, 58}} | {{harvnb|Gandz|1926|p=437}} }}</ref>}} The word entered the English language in the 16th century from [[Italian language|Italian]], [[Spanish language|Spanish]], and medieval [[Latin language|Latin]].<ref>{{multiref | {{harvnb|Cresswell|2010|p=[https://books.google.com/books?id=J4i3zV4vnBAC&pg=PA11 11]}} | {{harvnb|OUP Staff}} | {{harvnb|Menini|Oystaeyen|2017|p=722}} | {{harvnb|Hoad|1993|p=10}} }}</ref> Initially, its meaning was restricted to the [[theory of equations]], that is, to the art of manipulating [[polynomial equations]] in view of solving them. This changed in the 19th century{{efn|These changes were in part triggered by discoveries that solved many of the older problems of algebra. For example, the proof of the [[fundamental theorem of algebra]] demonstrated the existence of complex solutions of polynomials<ref name="auto7">{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|p=308}} | {{harvnb|Corry|2024|loc=§ The Fundamental Theorem of Algebra}} }}</ref> and the introduction of [[Galois theory]] characterized the polynomials that have [[solution in radicals|general solutions]].<ref name="auto2">{{multiref | {{harvnb|Kvasz|2006|pp=314–345}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Galois Theory, § Applications of Group Theory}} }}</ref>}} when the scope of algebra broadened to cover the study of diverse types of algebraic operations and structures together with their underlying [[axiom]]s, the laws they follow.<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Corry|2024|loc=§ Structural Algebra}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73–74]}} }}</ref>

== Major branches ==
=== Elementary algebra ===
{{main|Elementary algebra}}

[[File:algebraic equation notation.svg|thumb|right|class=skin-invert-image|alt=Diagram of an algebraic expression|Algebraic expression notation:<br />&nbsp; 1 – power (exponent)<br />&nbsp; 2 – coefficient<br />&nbsp; 3 – term<br />&nbsp; 4 – operator<br />&nbsp; 5 – constant term<br />&nbsp; <math>c</math> – constant<br />&nbsp; <math>x</math> <math>y</math> – variables]]

Elementary algebra, also called school algebra, college algebra, and classical algebra,<ref>{{multiref | {{harvnb|Arcavi|Drijvers|Stacey|2016|p=[https://books.google.com/books?id=N2R9DAAAQBAJ&pg=PA2 2]}} | {{harvnb|Benson|2003|pp=[https://books.google.com/books?id=nNbnCwAAQBAJ&pg=PA111 111–112]}} }}</ref> is the oldest and most basic form of algebra. It is a generalization of [[arithmetic]] that relies on [[Variable (mathematics)|variables]] and examines how mathematical [[Statement (logic)|statements]] may be transformed.<ref name="auto4">{{multiref | {{harvnb|Maddocks|2008|p=129}} | {{harvnb|Berggren|2015|loc=Lead section}} | {{harvnb|Pratt|2022|loc=§ 1. Elementary Algebra}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ 1. Historical Survey}} }}</ref> 

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of [[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]], [[exponentiation]], extraction of [[nth root|roots]], and [[logarithm]]. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in {{nowrap|<math>2 + 5 = 7</math>.<ref name="auto1"/>}}

Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables are [[symbol]]s for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the [[equation]] <math>2 \times 3 = 3 \times 2</math> belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the [[Commutative property|commutative property of multiplication]], which is expressed in the equation {{nowrap|<math>a \times b = b \times a</math>.<ref name="auto4"/>}}

[[Algebraic expression]]s are formed by using arithmetic operations to combine variables and numbers. By convention, the lowercase letters <math>x</math>, <math>y</math>, and <math>z</math> represent variables. In some cases, subscripts are added to distinguish variables, as in <math>x_1</math>, <math>x_2</math>, and <math>x_3</math>. The lowercase letters <math>a</math>, <math>b</math>, and <math>c</math> are usually used for [[Constant (mathematics)|constants]] and [[coefficient]]s.{{efn|Constants represent fixed numbers that do not change during the study of a specific problem.<ref>{{harvnb|Sobolev|2015}}</ref>}} The expression <math>5x + 3</math> is an algebraic expression created by multiplying the number 5 with the variable <math>x</math> and adding the number 3 to the result. Other examples of algebraic expressions are <math>32xyz</math> and {{nowrap|<math>64x_1^2 + 7x_2 - c</math>.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=129–130}} | {{harvnb|Young|2010|p=[https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA999 999]}} | {{harvnb|Majewski|2004|p=347}} | {{harvnb|Pratt|2022|loc=§ 1. Elementary Algebra}} | {{harvnb|Sorell|2000|p=[https://books.google.com/books?id=EksSDAAAQBAJ&pg=PA19 19]}} }}</ref>}}

Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is a statement formed by comparing two expressions, saying that they are equal. This can be expressed using the [[equals sign]] (<math>=</math>), as in {{nowrap|<math>5x^2 + 6x = 3y + 4</math>.}} [[Inequation]]s involve a different type of comparison, saying that the two sides are different. This can be expressed using symbols such as the [[less-than sign]] (<math><</math>), the [[greater-than sign]] (<math>></math>), and the inequality sign (<math>\neq</math>). Unlike other expressions, statements can be true or false, and their [[truth value]] usually depends on the values of the variables. For example, the statement <math>x^2 = 4</math> is true if <math>x</math> is either 2 or −2 and false otherwise.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=129–130}} | {{harvnb|Tsokos|Wooten|2015|p=[https://books.google.com/books?id=zu7HBQAAQBAJ&pg=PA451 451]}} | {{harvnb|Mishra|2016|p=1.2}} }}</ref> Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, such as the equation {{nowrap|<math>2x + 5x = 7x</math>.}} Conditional equations are only true for some values. For example, the equation <math>x + 4 = 9</math> is only true if <math>x</math> is 5.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|p=16}} | {{harvnb|Goodman|2001|p=[https://books.google.com/books?id=TvY7DQAAQBAJ&pg=PA5 5]}} | {{harvnb|Williams|2022}} }}</ref>

The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as [[Equation solving|solving the equation]] for that variable. For example, the equation <math>x - 7 = 4</math> can be solved for <math>x</math> by adding 7 to both sides, which isolates <math>x</math> on the left side and results in the equation {{nowrap|<math>x = 11</math>.<ref>{{multiref | {{harvnb|Maddocks|2008|p=130}} | {{harvnb|McKeague|1986|pp=[https://books.google.com/books?id=sq7iBQAAQBAJ&pg=PA51 51–54]}} | {{harvnb|Pratt|2022|loc=§ 1. Elementary Algebra}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ 1. Historical Survey}} }}</ref>}}

There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression <math>7x - 3x</math> can be replaced with the expression <math>4x</math> since <math>7x - 3x = (7-3)x = 4x</math> by the distributive property.<ref>{{multiref | {{harvnb|Tan|Steeb|Hardy|2012|p=[https://books.google.com/books?id=UDb0BwAAQBAJ&pg=PA306 306]}} | {{harvnb|Lamagna|2019|p=[https://books.google.com/books?id=8PSDDwAAQBAJ&pg=PA150 150]}} }}</ref> For statements with several variables, [[Substitution (logic)#Algebra|substitution]] is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that <math>y = 3x</math> then one can simplify the expression <math>7xy</math> to arrive at {{nowrap|<math>21x^2</math>.}} In a similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables.<ref>{{multiref | {{harvnb|Berggren|2015|loc=§ Solving Systems of Algebraic Equations}} | {{harvnb|McKeague|2014|p=[https://books.google.com/books?id=nI7iBQAAQBAJ&pg=PA386 386]}} | {{harvnb|McKeague|1986|p=[https://books.google.com/books?id=sq7iBQAAQBAJ&pg=PA148 148]}} }}</ref>

[[File:Graph (y = 0.5x - 1).svg|thumb|class=skin-invert-image|alt=Graph of equation "y = 0.5x − 1"|Algebraic equations can be used to describe geometric figures. All values for <math>x</math> and <math>y</math> that solve the equation are interpreted as points. They are drawn as a red, upward-sloping line in the graph above.]]

Algebraic equations can be interpreted [[Geometry|geometrically]] to describe spatial figures in the form of a [[Graph of a function|graph]]. To do so, the different variables in the equation are understood as [[Cartesian coordinate system|coordinates]] and the values that solve the equation are interpreted as points of a graph. For example, if <math>x</math> is set to zero in the equation <math>y=0.5x - 1</math>, then <math>y</math> must be −1 for the equation to be true. This means that the <math>(x, y)</math>-pair <math>(0, -1)</math> is part of the graph of the equation. The <math>(x, y)</math>-pair {{nowrap|<math>(0, 7)</math>,}} by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of <math>(x, y)</math>-pairs that solve the equation.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=130–131}} | {{harvnb|Rohde|Jain|Poddar|Ghosh|2012|p=[https://books.google.com/books?id=vk2XbZpsBOwC&pg=PT89 89]}} | {{harvnb|Walz|2016|loc=[https://www.spektrum.de/lexikon/mathematik/algebra/12062 Algebra]}}}}</ref>

==== Polynomials ====
{{main|Polynomial}}

A polynomial is an expression consisting of one or more terms that are added or subtracted from each other, like {{nowrap|<math>x^4+3xy^2+5x^3-1</math>.}} Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive integer power. A monomial is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials. The [[degree of a polynomial]] is the maximal value (among its terms) of the sum of the exponents of the variables (4 in the above example).<ref>{{multiref | {{harvnb|Bracken|Miller|2014|pp=386–387}} | {{harvnb|Kaufmann|Schwitters|2011|p=220}} | {{harvnb|Markushevich|2015}} }}</ref> Polynomials of degree one are called ''linear polynomials''. Linear algebra studies systems of linear polynomials.<ref>{{multiref | {{harvnb|Sahai|Bist|2002|p=[https://books.google.com/books?id=yjkgNYfApj4C&pg=PA21 21]}} | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Barrera-Mora|2023|pp=[https://books.google.com/books?id=Xmu8EAAAQBAJ&pg=PR9 ix, 1–2]}} }}</ref> A polynomial is said to be ''univariate'' or ''multivariate'', depending on whether it uses one or more variables.<ref>{{harvnb|Geddes|Czapor|Labahn|2007|p=[https://books.google.com/books?id=9fOUwkkRxT4C&pg=PA46 46]}}</ref>

[[Factorization]] is a method used to simplify polynomials, making it easier to analyze them and determine the values for which they [[Zero of a function|evaluate to zero]]. Factorization consists of rewriting a polynomial as a product of several factors. For example, the polynomial <math>x^2 - 3x - 10</math> can be factorized as {{nowrap|<math>(x + 2)(x - 5)</math>.}} The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if <math>x</math> is either −2 or 5.<ref>{{multiref | {{harvnb|Lukas|2022|pp=[https://books.google.com/books?id=1-dvEAAAQBAJ&pg=PA47 47–49]}} | {{harvnb|Berggren|2015|loc=§ Algebraic Expressions, § Solving Algebraic Equations}} }}</ref> Before the 19th century, much of algebra was devoted to [[polynomial equation]]s, that is [[equation]]s obtained by equating a polynomial to zero. The first attempts for solving polynomial equations were to express the solutions in terms of [[nth root|{{mvar|n}}th roots]]. The solution of a second-degree polynomial equation of the form <math>ax^2 + bx + c = 0</math> is given by the [[quadratic formula]]<ref>{{multiref | {{harvnb|Berggren|2015|loc=§ Solving algebraic equations}} | {{harvnb|Corry|2024|loc=§ Classical algebra}} }}</ref>
<math display=block>x = \frac{-b \pm \sqrt {b^2-4ac\ }}{2a}.</math>
Solutions for the degrees 3 and 4 are given by the [[cubic formula|cubic]] and [[quartic formula|quartic]] formulas. There are no general solutions for higher degrees, as proven in the 19th century by the [[Abel–Ruffini theorem]].<ref name="auto3"/> Even when general solutions do not exist, approximate solutions can be found by numerical tools like the [[Newton–Raphson method]].<ref>{{harvnb|Igarashi|Altman|Funada|Kamiyama|2014|p=[https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA103 103]}}</ref>

The [[fundamental theorem of algebra]] asserts that every univariate polynomial equation of positive degree with [[real number|real]] or [[complex number|complex]] coefficients has at least one complex solution. Consequently, every polynomial of a positive degree can be [[polynomial factorization|factorized]] into linear polynomials. This theorem was proved at the beginning of the 19th century, but this does not close the problem since the theorem does not provide any way for computing the solutions.<ref>{{multiref | {{harvnb|Berggren|2015|loc=§ Solving algebraic equations}} | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|p=308}} | {{harvnb|Corry|2024|loc=§ The Fundamental Theorem of Algebra}} }}</ref>

=== Linear algebra ===
{{main|Linear algebra}}

Linear algebra starts with the study of [[System of linear equations|systems of linear equations]].<ref>{{multiref | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Barrera-Mora|2023|pp=[https://books.google.com/books?id=Xmu8EAAAQBAJ&pg=PR9 ix, 1–2], }} }}</ref> An [[Linear equation|equation is linear]] if it can be expressed in the form <math>a_1x_1 + a_2x_2 + ... + a_nx_n = b</math> where <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math> and <math>b</math> are constants. Examples are <math>x_1 - 7x_2 + 3x_3 = 0</math> and <math display=inline>\frac14x - y = 4</math>. A ''system of linear equations'' is a set of linear equations for which one is interested in common solutions.<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=2–3}} | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Voitsekhovskii|2011}} }}</ref>

[[Matrix (mathematics)|Matrices]] are rectangular arrays of values that have been originally introduced for having a compact and synthetic notation for systems of linear equations.<ref>{{multiref | {{harvnb|Saikia|2008|p=[https://books.google.com/books?id=KhM7BAAAQBAJ&pg=PA1 1]}} | {{harvnb|Lal|2017|p=[https://books.google.com/books?id=FwPNDgAAQBAJ&pg=PA31 31]}} | {{harvnb|Mirakhor|Krichene|2014|p=[https://books.google.com/books?id=G6XNAwAAQBAJ&pg=PT107 107]}} }}</ref> For example, the system of equations
<math display="block"> \begin{align}
 9x_1 + 3x_2 - 13x_3 &= 0 \\
 2.3x_1 + 7x_3 &= 9 \\
 -5x_1 - 17x_2 &= -3
\end{align}
</math>
can be written as 
<math display="block">AX=B,</math>
where <math>A,X</math> and <math>B</math> are the matrices
<math display="block">A=\begin{bmatrix}9 & 3 & -13 \\ 2.3 & 0 & 7 \\ -5 & -17 & 0 \end{bmatrix}, \quad X= \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix},\quad B = \begin{bmatrix}0 \\ 9 \\ -3 \end{bmatrix}.</math>

Under some conditions on the number of rows and columns, matrices can be [[matrix addition|added]], [[matrix multiplication|multiplied]], and sometimes [[inverse matrix|inverted]]. All methods for solving linear systems may be expressed as matrix manipulations using these operations. For example, solving the above system consists of computing an inverted matrix <math>A^{-1}</math> such that <math>A^{-1}A = I,</math> where <math>I</math> is the [[identity matrix]]. Then, multiplying on the left both members of the above matrix equation by <math>A^{-1},</math> one gets the solution of the system of linear equations as<ref>{{multiref | {{harvnb|Brown|2015|pp=[https://books.google.com/books?id=POlVBgAAQBAJ&pg=PA30 30–31]}} | {{harvnb|Waerden|2003|pp=70–72}} }}</ref>
<math display="block">X=A^{-1}B.</math>

Methods of solving systems of linear equations range from the introductory, like substitution<ref>{{multiref | {{harvnb|Young|2010|pp=[https://books.google.com/books?id=9HRLAn326zEC&pg=PA697 697–698]}} | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Sullivan|2010|pp=[https://books.google.com/books?id=6NKaDwAAQBAJ&pg=PA53 53–54]}} }}</ref> and elimination,<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=7–8}} | {{harvnb|Sullivan|2010|pp=[https://books.google.com/books?id=6NKaDwAAQBAJ&pg=PA55 55–56]}} | {{harvnb|Atanasiu|Mikusinski|2019|p=[https://books.google.com/books?id=VbySDwAAQBAJ&pg=PA75 75]}}}}</ref> to more advanced techniques using matrices, such as [[Cramer's rule]], the [[Gaussian elimination]], and [[LU decomposition]].<ref>{{multiref | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Anton|Rorres|2013|pp=7–8, 11, 491}} }}</ref> Some systems of equations are [[Consistent and inconsistent equations|inconsistent]], meaning that no solutions exist because the equations contradict each other.<ref name="auto10">{{multiref | {{harvnb|Anton|Rorres|2013|pp=3–7}} | {{harvnb|Mortensen|2013|pp=[https://books.google.com/books?id=KYDrCAAAQBAJ&pg=PA73 73–74]}} | {{harvnb|Young|2023|pp=[https://books.google.com/books?id=pMSZEAAAQBAJ&pg=PA714 714–715]}} }}</ref>{{efn|For example, the equations <math>x_1 - 3x_2 = 0</math> and <math>x_1 - 3x_2 = 7</math> contradict each other since no values of <math>x_1</math> and <math>x_2</math> exist that solve both equations at the same time.<ref name="auto10"/>}} Consistent systems have either one unique solution or an infinite number of solutions.<ref name="auto11">{{multiref | {{harvnb|Maddocks|2008|p=131}} | {{harvnb|Harrison|Waldron|2011|p=[https://books.google.com/books?id=_sisAgAAQBAJ&pg=PT464 464]}} | {{harvnb|Anton|2013|p=[https://books.google.com/books?id=neYGCwAAQBAJ&pg=PA255 255]}} }}</ref>{{efn|Whether a consistent system of equations has a unique solution depends on the number of variables and [[independent equation]]s. Several equations are independent of each other if they do not provide the same information and cannot be derived from each other. A unique solution exists if the number of variables is the same as the number of independent equations. [[Underdetermined system]]s, by contrast, have more variables than independent equations and have an infinite number of solutions if they are consistent.<ref name="auto11"/>}}

The study of [[vector space]]s and [[linear map]]s form a large part of linear algebra. A vector space is an algebraic structure formed by a set with an addition that makes it an [[abelian group]] and a [[scalar multiplication]] that is compatible with addition (see [[vector space]] for details). A linear map is a function between vector spaces that is compatible with addition and scalar multiplication. In the case of [[finite-dimensional vector space]]s, vectors and linear maps can be represented by matrices. It follows that the theories of matrices and finite-dimensional vector spaces are essentially the same. In particular, vector spaces provide a third way for expressing and manipulating systems of linear equations.<ref>{{multiref | {{harvnb|Valenza|2012|p=[https://books.google.com/books?id=7x8MCAAAQBAJ&pg=PR7 vii]}} | {{harvnb|Chahal|2018|loc=[https://books.google.com/books?id=BGR8DwAAQBAJ&pg=PT10 § 1.1 What is Linear Algebra?]}} | {{harvnb|Solomon|2014|pp=[https://books.google.com/books?id=BpvSBQAAQBAJ&pg=PA57 57–58, 61–62]}} | {{harvnb|Ricardo|2009|p=[https://books.google.com/books?id=s7bMBQAAQBAJ&pg=PA389 389]}} }}</ref> From this perspective, a matrix is a representation of a linear map: if one chooses a particular [[Basis (linear algebra)|basis]] to describe the vectors being transformed, then the entries in the matrix give the results of applying the linear map to the basis vectors.<ref>{{multiref | {{harvnb|Solomon|2014|p=[https://books.google.com/books?id=BpvSBQAAQBAJ&pg=PA57 57]}} | {{harvnb|Ricardo|2009|pp=[https://books.google.com/books?id=s7bMBQAAQBAJ&pg=PA395 395–396]}} }}</ref>

[[File:Linear Function Graph.svg|thumb|alt=Graph of two linear equations|Linear equations with two variables can be interpreted geometrically as lines. The solution of a system of linear equations is where the lines intersect.]]

Systems of equations can be interpreted as geometric figures. For systems with two variables, each equation represents a [[Line (geometry)|line]] in [[two-dimensional space]]. The point where the two lines intersect is the solution of the full system because this is the only point that solves both the first and the second equation. For inconsistent systems, the two lines run parallel, meaning that there is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to seek solutions graphically by plotting the equations and determining where they intersect.<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=3–5}} | {{harvnb|Young|2010|pp=[https://books.google.com/books?id=9HRLAn326zEC&pg=PA696 696–697]}} | {{harvnb|Sneyd|Fewster|McGillivray|2022|p=[https://books.google.com/books?id=zqd3EAAAQBAJ&pg=PA211 211]}} }}</ref> The same principles also apply to systems of equations with more variables, with the difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to [[Plane (mathematics)|planes]] in [[three-dimensional space]], and the points where all planes intersect solve the system of equations.<ref>{{multiref | {{harvnb|Anton|Rorres|2013|pp=3–5}} | {{harvnb|Young|2010|p=[https://books.google.com/books?id=9HRLAn326zEC&pg=PA713 713]}} | {{harvnb|Sneyd|Fewster|McGillivray|2022|p=[https://books.google.com/books?id=zqd3EAAAQBAJ&pg=PA211 211]}} }}</ref>

=== Abstract algebra ===
{{main|Abstract algebra}}

Abstract algebra, also called modern algebra,<ref>{{multiref | {{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA1 1]}} | {{harvnb|Dominich|2008|p=[https://books.google.com/books?id=uEedNKV3nlUC&pg=PA19 19]}} }}</ref> is the study of [[algebraic structures]]. An algebraic structure is a framework for understanding [[Operation (mathematics)|operations]] on [[mathematical object]]s, like the addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].<ref>{{multiref | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead section, § 2. Abstract Algebra}} | {{harvnb|Gilbert|Nicholson|2004|pp=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA1 1–3]}} | {{harvnb|Dominich|2008|p=[https://books.google.com/books?id=uEedNKV3nlUC&pg=PA19 19]}} }}</ref> The key difference between these types of algebraic structures lies in the number of operations they use and the laws they obey.<ref name="auto9"/> In [[mathematics education]], abstract algebra refers to an advanced [[undergraduate]] course that mathematics majors take after completing courses in linear algebra.<ref>{{harvnb|Hausberger|2020|loc=Abstract Algebra Teaching and Learning}}</ref>

[[File:Binary operations as black box.svg|thumb|class=skin-invert-image|alt=Diagram of binary operation|Many algebraic structures rely on binary operations, which take two objects as their input and combine them into a single object as output, like addition and multiplication do.]]

On a formal level, an algebraic structure is a [[Set (mathematics)|set]]{{efn|A set is an unordered collection of distinct elements, such as numbers, vectors, or other sets. [[Set theory]] describes the laws and properties of sets.<ref>{{multiref | {{harvnb|Tanton|2005|p=460}} | {{harvnb|Murthy|2012|p=[https://books.google.com/books?id=CBc8BAAAQBAJ&pg=PA3 1.3]}} }}</ref>}} of mathematical objects, called the underlying set, together with one or several operations.{{efn|According to some definitions, algebraic structures include a distinguished element as an additional component, such as the identity element in the case of multiplication.<ref name="auto5">{{harvnb|Ovchinnikov|2015|p=[https://books.google.com/books?id=UMbXBgAAQBAJ&pg=PA27 27]}}</ref>}} Abstract algebra is primarily interested in [[binary operation]]s,{{efn|Some of the algebraic structures studied by abstract algebra include [[unary operation]]s in addition to binary operations. For example, [[normed vector space]]s have a [[Norm (mathematics)|norm]], which is a unary operation often used to associate a vector with its length.<ref>{{harvnb|Grillet|2007|p=[https://books.google.com/books?id=LJtyhu8-xYwC&pg=PA247 247]}}</ref>}} which take any two objects from the underlying set as inputs and map them to another object from this set as output.<ref>{{multiref | {{harvnb|Whitelaw|1995|p=[https://books.google.com/books?id=f2hyf0QoB_0C&pg=PA61 61]}} | {{harvnb|Nicholson|2012|p=[https://books.google.com/books?id=w-GaLpapRcEC&pg=PA70 70]}} | {{harvnb|Fiche|Hebuterne|2013|p=[https://books.google.com/books?id=TqkckiuuXg8C&pg=PT326 326]}} | {{harvnb|Pratt|2022|loc=Lead section, § 2. Abstract Algebra}} }}</ref> For example, the algebraic structure <math>\langle \N, + \rangle</math> has the [[natural numbers]] (<math>\N</math>) as the underlying set and addition (<math>+</math>) as its binary operation.<ref name="auto5"/> The underlying set can contain mathematical objects other than numbers, and the operations are not restricted to regular arithmetic operations.<ref>{{multiref | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead section, § 2. Abstract Algebra}} }}</ref> For instance, the underlying set of the [[symmetry group]] of a geometric object is made up of [[geometric transformation]]s, such as [[rotation]]s, under which the object remains [[Invariant (mathematics)|unchanged]]. Its binary operation is [[function composition]], which takes two transformations as input and has the transformation resulting from applying the first transformation followed by the second as its output.<ref>{{multiref | {{harvnb|Olver|1999|pp=[https://books.google.com/books?id=1GlHYhNRAqEC&pg=PA55 55–56]}} | {{harvnb|Abas|Salman|1994|pp=[https://books.google.com/books?id=5snsCgAAQBAJ&pg=PA58 58–59]}} | {{harvnb|Häberle|2009|p=[https://books.google.com/books?id=McvSa-cFZCMC&pg=PA640 640]}} }}</ref>

====Group theory====
{{main|Group theory}}
Abstract algebra classifies algebraic structures based on the laws or [[axiom]]s that its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation is [[Associative property|associative]] and has an [[identity element]] and [[inverse element]]s. An operation is associative if the order of several applications does not matter, i.e., if {{nowrap|<math>(a \circ b) \circ c</math>{{efn|The symbols <math>\circ</math> and <math>\star</math> are used in this article to represent any operation that may or may not resemble arithmetic operations.<ref>{{harvnb|Gilbert|Nicholson|2004|p=[https://books.google.com/books?id=paINAXYHN8kC&pg=PA4 4]}}</ref>}}}} is the same as <math>a \circ (b \circ c)</math> for all elements. An operation has an identity element or a neutral element if one element ''e'' exists that does not change the value of any other element, i.e., if {{nowrap|<math>a \circ e = e \circ a = a</math>.}} An operation has inverse elements if for any element <math>a</math> there exists a reciprocal element <math>a^{-1}</math> that undoes <math>a</math>. If an element operates on its inverse then the result is the neutral element ''e'', expressed formally as {{nowrap|<math>a \circ a^{-1} = a^{-1} \circ a = e</math>.}} Every algebraic structure that fulfills these requirements is a group.<ref>{{multiref | {{harvnb|Kargapolov|Merzlyakov|2016|loc=§ Definition}} | {{harvnb|Khattar|Agrawal|2023|pp=[https://books.google.com/books?id=7-nIEAAAQBAJ&pg=PA4 4–6]}} | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Pratt|2022|loc=Lead section, § 2. Abstract Algebra}} | {{harvnb|Neri|2019|p=[https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PA258 258]}} }}</ref> For example, <math>\langle \Z, + \rangle</math> is a group formed by the set of [[integers]] together with the operation of addition. The neutral element is 0 and the inverse element of any number <math>a</math> is {{nowrap|<math>-a</math>.<ref>{{multiref | {{harvnb|Khattar|Agrawal|2023|pp=[https://books.google.com/books?id=7-nIEAAAQBAJ&pg=PA6 6–7]}} | {{harvnb|Maddocks|2008|pp=131–132}} | {{harvnb|Adhikari|Adhikari|2013|p=[https://books.google.com/books?id=lBO7BAAAQBAJ&pg=PA72 72]}}}}</ref>}} The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements.<ref>{{multiref | {{harvnb|McWeeny|2002|p=[https://books.google.com/books?id=x3fjIXY93TsC&pg=PA6 6]}} | {{harvnb|Kramer|Pippich|2017|p=[https://books.google.com/books?id=nvM-DwAAQBAJ&pg=PA49 49]}} }}</ref>

[[Group theory]] examines the nature of groups, with basic theorems such as the [[fundamental theorem of finite abelian groups]] and the [[Feit–Thompson theorem]].<ref>{{multiref | {{harvnb|Tanton|2005|p=242}} | {{harvnb|Bhattacharya|Jain|Nagpaul|1994|p=[https://books.google.com/books?id=hiQ8e0b48swC&pg=PA141 141]}} | {{harvnb|Weisstein|2003|p=[https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA1020 1020]}} }}</ref> The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete [[classification of finite simple groups]].<ref>{{multiref | {{harvnb|Elwes|2006}} | {{harvnb|Wilson|2009|p=2}} }}</ref>

====Ring theory and field theory====
{{main|Ring theory|Field (mathematics)}}
A ring is an algebraic structure with two operations that work similarly to the addition and multiplication of numbers and are named and generally denoted similarly. A ring is a [[commutative group]] under addition: the addition of the ring is associative, commutative, and has an identity element and inverse elements. The multiplication is associative and [[distributivity|distributive]] with respect to addition; that is, <math>a (b + c) = a b + a c</math> and <math>(b + c) a = b a + c a.</math> Moreover, multiplication is associative and has an [[identity element]] generally denoted as {{math|1}}.<ref>{{multiref | {{harvnb|Weisstein|2003|p=2579}} | {{harvnb|Maxwell|2009|pp=[https://books.google.com/books?id=yD0irRUE_u4C&pg=PA73 73–74]}} | {{harvnb|Pratt|2022|loc=§ 2.3 Rings}} }}</ref>{{efn|Some authors do not require the existence of multiplicative identity elements. A ring without multiplicative identity is sometimes called a {{not a typo|[[rng (algebra)|rng]].}}<ref>{{harvnb|Silverman|2022|p=[https://books.google.com/books?id=PKFnEAAAQBAJ&pg=PA64 64]}}</ref>}} Multiplication needs not to be commutative; if it is commutative, one has a [[commutative ring]].<ref>{{harvnb|Geddes|Czapor|Labahn|2007|p=[https://books.google.com/books?id=9fOUwkkRxT4C&pg=PA24 24]}}</ref> The [[ring of integers]] (<math>\Z</math>) is one of the simplest commutative rings.<ref>{{harvnb|Smith|2015|p=[https://books.google.com/books?id=MXu9CgAAQBAJ&pg=PA161 161]}}</ref>

A ''[[field (mathematics)|field]]'' is a commutative ring such that {{tmath|1\ne 0}} and each nonzero element has a [[multiplicative inverse]].<ref>{{multiref | {{harvnb|Geddes|Czapor|Labahn|2007|p=[https://books.google.com/books?id=9fOUwkkRxT4C&pg=PA24 24]}} | {{harvnb|Weisstein|2003|pp=1047, 2579}} | {{harvnb|Pratt|2022|loc=§ 2.4 Fields}} }}</ref> The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of <math>7</math> is {{nowrap|<math>\tfrac{1}{7}</math>,}} which is not an integer. The [[rational numbers]], the [[real numbers]], and the [[complex numbers]] each form a field with the operations of addition and multiplication.<ref>{{multiref | {{harvnb|Irving|2004|pp=77, 236}} | {{harvnb|Weisstein|2003|pp=1047, 2579}} | {{harvnb|Hohn|2013|pp=[https://books.google.com/books?id=9XlFu4XQ6nUC&pg=PA83 83–84]}} }}</ref>

[[Ring theory]] is the study of rings, exploring concepts such as [[subring]]s, [[quotient ring]]s, [[polynomial ring]]s, and [[Ideal (ring theory)|ideals]] as well as theorems such as [[Hilbert's basis theorem]].<ref>{{multiref | {{harvnb|Serovajsky|2020|loc=§ Room 4B.5 Rings}} | {{harvnb|Kleiner|2007|p=[https://books.google.com/books?id=udj-1UuaOiIC&pg=PA63 63]}} | {{harvnb|Kline|1990|p=[https://books.google.com/books?id=8YaBuGcmLb0C&pg=PA1153 1153]}} }}</ref> Field theory is concerned with fields, examining [[field extension]]s, [[algebraic closure]]s, and [[finite fields]].<ref>{{multiref | {{harvnb|Waerden|2003|pp=110–114, 231, 246}} | {{harvnb|Karpilovsky|1989|p=[https://books.google.com/books?id=-2WR0fw9gLMC&pg=PA45 45]}} | {{harvnb|Kleiner|2007|p=[https://books.google.com/books?id=udj-1UuaOiIC&pg=PA63 63]}} }}</ref> [[Galois theory]] explores the relation between field theory and group theory, relying on the [[fundamental theorem of Galois theory]].<ref>{{multiref | {{harvnb|Lang|2005|pp=[https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA261 261–262]}} | {{harvnb|Cox|2012|pp=[https://books.google.com/books?id=vBKrOch1AkYC&pg=PA161 161–162]}} }}</ref>

====Theories of interrelations among structures====

[[File:Magma to group4.svg|thumb|class=skin-invert-image|alt=Diagram of relations between some algebraic structures|Diagram of relations between some algebraic structures. For instance, its top right section shows that a [[Magma (algebra)|magma]] becomes a [[semigroup]] if its operation is associative.]]

Besides groups, rings, and fields, there are many other algebraic structures studied by algebra. They include [[Magma (algebra)|magmas]], [[semigroup]]s, [[monoid]]s, [[abelian group]]s, [[commutative ring]]s, [[Module (mathematics)|modules]], [[Lattice (order)|lattices]], [[vector space]]s, [[Algebra over a field|algebras over a field]], and [[associative]] and [[non-associative algebras]]. They differ from each other regarding the types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements.<ref name="auto9">{{multiref | {{harvnb|Pratt|2022|loc=Lead section, § 2. Abstract Algebra}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.}} | {{harvnb|Bourbaki|1998|pp=[https://books.google.com/books?id=STS9aZ6F204C&pg=PA428 428–430, 446]}} }}</ref> For example, a magma becomes a semigroup if its operation is associative.<ref>{{harvnb|Cooper|2011|p=[https://books.google.com/books?id=Fybzl6QB62gC&pg=PA60 60]}}</ref>

[[Homomorphism]]s are tools to examine structural features by comparing two algebraic structures.<ref>{{multiref | {{harvnb|Rowen|2006|p=[https://books.google.com/books?id=AhEPCgAAQBAJ&pg=PA12 12]}} | {{harvnb|Pratt|2022|loc=§ 3.3 Birkhoff's Theorem}} | {{harvnb|Grätzer|2008|p=34}} }}</ref> A homomorphism is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form <math>\langle A, \circ \rangle</math> and <math>\langle B, \star \rangle</math> then the function <math>h: A \to B</math> is a homomorphism if it fulfills the following requirement: {{nowrap|<math>h(x \circ y) = h(x) \star h(y)</math>.}} The existence of a homomorphism reveals that the operation <math>\star</math> in the second algebraic structure plays the same role as the operation <math>\circ</math> does in the first algebraic structure.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3.3 Birkhoff's Theorem}} | {{harvnb|Rowen|2006|p=[https://books.google.com/books?id=AhEPCgAAQBAJ&pg=PA12 12]}} | {{harvnb|Gowers|Barrow-Green|Leader|2010|pp=[https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA27 27–28]}} | {{harvnb|Adhikari|2016|pp=[https://books.google.com/books?id=MIAXDQAAQBAJ&pg=PA5 5–6]}} }}</ref> [[Isomorphisms]] are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a [[bijective]] homomorphism, meaning that it establishes a one-to-one relationship between the elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure.<ref>{{multiref | {{harvnb|Neri|2019|pp=[https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PA278 278–279]}} | {{harvnb|Ivanova|Smirnov|2012}} | {{harvnb|Deo|2018|p=[https://books.google.com/books?id=HjtRDwAAQBAJ&pg=PA295 295]}} | {{harvnb|Ono|2019|p=[https://books.google.com/books?id=SR2nDwAAQBAJ&pg=PA84 84]}}}}</ref>

[[File:Venn A subset B.svg|thumb|alt=Venn diagram of a set and its subset|[[Subalgebra]]s restrict their operations to a subset of the underlying set of the original algebraic structure.]]

Another tool of comparison is the relation between an algebraic structure and its [[Subalgebra#Subalgebras in universal algebra|subalgebra]].<ref name="auto8">{{multiref | {{harvnb|Indurkhya|2013|pp=[https://books.google.com/books?id=foTrCAAAQBAJ&pg=PA217 217–218]}} | {{harvnb|Pratt|2022|loc=§ 3.3 Birkhoff's Theorem}} | {{harvnb|Grätzer|2008|p=34}} }}</ref> The algebraic structure and its subalgebra use the same operations,{{efn|According to some definitions, it is also possible for a subalgebra to have fewer operations.<ref name="auto6">{{harvnb|Indurkhya|2013|pp=[https://books.google.com/books?id=foTrCAAAQBAJ&pg=PA217 217–218]}}</ref>}} which follow the same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure.{{efn|This means that all the elements of the first set are also elements of the second set but the second set may contain elements not found in the first set.<ref>{{harvnb|Efimov|2014}}</ref>}} All operations in the subalgebra are required to be [[Closure (mathematics)|closed]] in its underlying set, meaning that they only produce elements that belong to this set.<ref name="auto8"/> For example, the set of [[Parity (mathematics)|even integers]] together with addition is a subalgebra of the full set of integers together with addition. This is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra because it is not closed: adding two odd numbers produces an even number, which is not part of the chosen subset.<ref name="auto6"/>

[[Universal algebra]] is the study of algebraic structures in general. As part of its general perspective, it is not concerned with the specific elements that make up the underlying sets and considers operations with more than two inputs, such as [[ternary operation]]s. It provides a framework for investigating what structural features different algebraic structures have in common.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3. Universal Algebra}} | {{harvnb|Cohn|2012|p=[https://books.google.com/books?id=6tbuCAAAQBAJ&pg=PR13 xiii]}} }}</ref>{{efn|A slightly different approach understands universal algebra as the study of one type of algebraic structures known as universal algebras. Universal algebras are defined in a general manner to include most other algebraic structures. For example, groups and rings are special types of universal algebras.<ref>{{multiref | {{harvnb|Smirnov|2020}} | {{harvnb|Grätzer|2008|pp=7–8}} | {{harvnb|Bahturin|2013|p=[https://books.google.com/books?id=8RbvCAAAQBAJ&pg=PA346 346]}} }}</ref>}} One of those structural features concerns the [[Identity (mathematics)#Logic and universal algebra|identities]] that are true in different algebraic structures. In this context, an identity is a [[Universal quantification|universal]] equation or an equation that is true for all elements of the underlying set. For example, commutativity is a universal equation that states that <math>a \circ b</math> is identical to <math>b \circ a</math> for all elements.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 3.2 Equational Logic}} | {{harvnb|Mal’cev|1973|pp=210–211}} }}</ref> A [[Variety (universal algebra)|variety]] is a class of all algebraic structures that satisfy certain identities. For example, if two algebraic structures satisfy commutativity then they are both part of the corresponding variety.<ref>{{multiref | {{harvnb|Mal’cev|1973|pp=210–211}} | {{harvnb|Cohn|2012|p=[https://books.google.com/books?id=6tbuCAAAQBAJ&pg=PA162 162]}} | {{harvnb|Rosen|2012|p=[https://books.google.com/books?id=-oVvEAAAQBAJ&pg=PA779 779]}} | {{harvnb|Hazewinkel|1994|p=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA406 406]}} }}</ref>{{efn|Not every type of algebraic structure forms a variety. For example, both groups and rings form varieties but fields do not.<ref>{{harvnb|Cohn|1995|p=[https://books.google.com/books?id=u-4ADgUgpSMC&pg=PA8 8]}}</ref>}}{{efn|Besides identities, universal algebra is also interested in structural features associated with [[quasi-identity|quasi-identities]]. A quasi-identity is an identity that only needs to be present under certain conditions (which take the form of a [[Horn clause]]<ref>{{harvnb|Mal’cev|1973|p=211}}</ref>). It is a generalization of identity in the sense that every identity is a quasi-identity but not every quasi-identity is an identity. A [[quasivariety]] is a class of all algebraic structures that satisfy certain quasi-identities.<ref>{{multiref | {{harvnb|Mal’cev|1973|pp=210–211}} | {{harvnb|Pratt|2022|loc=§ 3. Universal Algebra}} | {{harvnb|Artamonov|2003|p=[https://books.google.com/books?id=sLDGY4Hk8V0C&pg=PA873 873]}} }}</ref>}}

[[Category theory]] examines how mathematical objects are related to each other using the concept of [[Category (mathematics)|categories]]. A category is a collection of objects together with a collection of [[morphisms]] or "arrows" between those objects. These two collections must satisfy certain conditions. For example, morphisms can be joined, or ''composed'': if there exists a morphism from object <math>a</math> to object <math>b</math>, and another morphism from object <math>b</math> to object <math>c</math>, then there must also exist one from object <math>a</math> to object <math>c</math>. Composition of morphisms is required to be associative, and there must be an "identity morphism" for every object.<ref>{{multiref | {{harvnb|Weisstein|2003|pp=347–348}} | {{harvnb|Gowers|Barrow-Green|Leader|2010|pp=[https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA6 6, 165]}} | {{harvnb|Cheng|2023|p=102}} }}</ref> Categories are widely used in contemporary mathematics since they provide a unifying framework to describe and analyze many fundamental mathematical concepts. For example, sets can be described with the [[category of sets]], and any group can be regarded as the morphisms of a category with just one object.<ref>{{multiref | {{harvnb|Gowers|Barrow-Green|Leader|2010|pp=[https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA6 6, 165]}} | {{harvnb|Borceux|1994|p=[https://books.google.com/books?id=YfzImoopB-IC&pg=PA20 20]}} | {{harvnb|Laos|1998|p=[https://books.google.com/books?id=1r7dSn4ZqogC&pg=PA100 100]}} | {{harvnb|Cheng|2023|pp=128–131}} }}</ref>

== History ==
{{main|History of algebra|Timeline of algebra}}

[[File:Rhind Mathematical Papyrus.jpg|thumb|alt=Rhind Papyrus|The [[Rhind Mathematical Papyrus]] from [[ancient Egypt]], dated {{circa|1650 BCE}}, is one of the earliest documents discussing algebraic problems.]]

The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities. These developments happened in the ancient period in [[Babylonia]], [[Ancient Egypt|Egypt]], [[Ancient Greece|Greece]], [[Ancient China|China]], and [[Ancient India|India]]. One of the earliest documents on algebraic problems is the [[Rhind Mathematical Papyrus]] from ancient Egypt, which was written around 1650&nbsp;BCE.{{efn|The exact date is disputed and some historians suggest a later date around 1550 BCE.<ref>{{multiref | {{harvnb|Corry|2024|loc=§ Problem-Solving in Egypt and Babylon}} | {{harvnb|Brezinski|Meurant|Redivo-Zaglia|2022|p=[https://books.google.com/books?id=4IGhEAAAQBAJ&pg=PA34 34]}} }}</ref>}} It discusses solutions to [[linear equations]], as expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?" Babylonian clay tablets from around the same time explain methods to solve linear and [[Quadratic equation|quadratic polynomial equations]], such as the method of [[completing the square]].<ref>{{multiref | {{harvnb|Tanton|2005|p=9}} | {{harvnb|Kvasz|2006|p=290}} | {{harvnb|Corry|2024|loc=§ Problem Solving in Egypt and Babylon}} }}</ref>

Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main interest was geometry rather than algebra, but they employed algebraic methods to solve geometric problems. For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified in [[Pythagoras]]' formulation of the [[difference of two squares]] method and later in [[Euclid's Elements|Euclid's ''Elements'']].<ref>{{multiref | {{harvnb|Tanton|2005|p=9}} | {{harvnb|Kvasz|2006|p=290}} | {{harvnb|Corry|2024|loc=§ The Pythagoreans and Euclid}} }}</ref> In the 3rd century CE, [[Diophantus]] provided a detailed treatment of how to solve algebraic equations in a series of books called ''[[Arithmetica]]''. He was the first to experiment with symbolic notation to express polynomials.<ref>{{multiref | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} | {{harvnb|Sialaros|2018|p=[https://books.google.com/books?id=2PZYDwAAQBAJ&pg=PT55 55]}} | {{harvnb|Corry|2024|loc=§ Diophantus}} }}</ref> Diophantus's work influenced Arab development of algebra with many of his methods reflected in the concepts and techniques used in medieval Arabic algebra.<ref>{{multiref | {{harvnb|Hettle|2015|pp=139–141, 160–161}} | {{harvnb|Christianidis|Megremi|2019|pp=16–17}} }}</ref> In ancient China, ''[[The Nine Chapters on the Mathematical Art]]'', a book composed over the period spanning from the 10th century BCE to the 2nd century CE,<ref>{{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA10 10]}}</ref> explored various techniques for solving algebraic equations, including the use of matrix-like constructs.<ref>{{harvnb|Higgins|2015|p=[https://books.google.com/books?id=QANiCgAAQBAJ&pg=PA89 89]}}</ref>

There is no unanimity as to whether these early developments are part of algebra or only precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications.<ref>{{multiref | {{harvnb|Kvasz|2006|pp=290–291}} | {{harvnb|Sialaros|2018|p=[https://books.google.com/books?id=2PZYDwAAQBAJ&pg=PT55 55]}} | {{harvnb|Boyer|Merzbach|2011|p=[https://books.google.com/books?id=bR9HAAAAQBAJ&pg=PA161 161]}} | {{harvnb|Derbyshire|2006|p=[https://books.google.com/books?id=mLqaAgAAQBAJ&pg=PT39 31]}} }}</ref> This changed with the Persian mathematician [[al-Khwarizmi]],{{efn|Some historians consider him the "father of algebra" while others reserve this title for Diophantus.<ref>{{multiref | {{harvnb|Boyer|Merzbach|2011|p=[https://books.google.com/books?id=bR9HAAAAQBAJ&pg=PA161 161]}} | {{harvnb|Derbyshire|2006|p=[https://books.google.com/books?id=mLqaAgAAQBAJ&pg=PA31 31]}} }}</ref> }} who published his ''[[The Compendious Book on Calculation by Completion and Balancing]]'' in 825 CE. It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides.<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|pp=291–293}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} }}</ref> Other influential contributions to algebra came from the Arab mathematician [[Thābit ibn Qurra]] also in the 9th century and the Persian mathematician [[Omar Khayyam]] in the 11th and 12th centuries.<ref>{{multiref | {{harvnb|Waerden|2013|pp=3, 15–16, 24–25}} | {{harvnb|Jenkins|2010|p=[https://books.google.com/books?id=giEkCQAAQBAJ&pg=PA82 82]}} | {{harvnb|Pickover|2009|p=[https://books.google.com/books?id=JrslMKTgSZwC&pg=PA90 90]}} }}</ref>

In India, [[Brahmagupta]] investigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his innovations were the use of zero and negative numbers in algebraic equations.<ref>{{multiref | {{harvnb|Tanton|2005|pp=9–10}} | {{harvnb|Corry|2024|loc=§ The Equation in India and China}} }}</ref> The Indian mathematicians [[Mahāvīra (mathematician)|Mahāvīra]] in the 9th century and [[Bhāskara II]] in the 12th century further refined Brahmagupta's methods and concepts.<ref>{{multiref | {{harvnb|Seshadri|2010|p=[https://books.google.com/books?id=w_JdDwAAQBAJ&pg=PA156 156]}} | {{harvnb|Emch|Sridharan|Srinivas|2005|p=[https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA20 20]}} }}</ref> In 1247, the Chinese mathematician [[Qin Jiushao]] wrote the ''[[Mathematical Treatise in Nine Sections]]'', which includes [[Horner's method|an algorithm]] for the [[Polynomial evaluation|numerical evaluation of polynomials]], including polynomials of higher degrees.<ref>{{multiref | {{harvnb|Smorynski|2007|p=[https://books.google.com/books?id=qY657eFq7UgC&pg=PA137 137]}} | {{harvnb|Zwillinger|2002|p=[https://books.google.com/books?id=gE_MBQAAQBAJ&pg=PA812 812]}} }}</ref>

{{multiple image
|perrow=2
|total_width=350
|image1=Francois Viete.jpeg
|alt1=Drawing of François Viète
|image2=Frans Hals - Portret van René Descartes (cropped).jpg
|alt2=Painting of René Descartes
|footer=[[François Viète]] (left) and [[René Descartes]] invented a symbolic notation to express equations in an abstract and concise manner.
}}

The Italian mathematician [[Fibonacci]] brought al-Khwarizmi's ideas and techniques to Europe in books including his ''[[Liber Abaci]]''.<ref>{{multiref | {{harvnb|Waerden|2013|pp=32–35}} | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|p=293}} }}</ref> In 1545, the Italian polymath [[Gerolamo Cardano]] published his book ''[[Ars Magna (Cardano book)|Ars Magna]]'', which covered many topics in algebra, discussed [[imaginary numbers]], and was the first to present general methods for solving [[Cubic equation|cubic]] and [[quartic equation]]s.<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|p=293}} | {{harvnb|Corry|2024|loc=§ Cardano and the Solving of Cubic and Quartic Equations}} | {{harvnb|Miyake|2002|p=[https://books.google.com/books?id=G0P2BwAAQBAJ&pg=PA268 268]}} }}</ref> In the 16th and 17th centuries, the French mathematicians [[François Viète]] and [[René Descartes]] introduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner. Their predecessors had relied on verbal descriptions of problems and solutions.<ref>{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Kvasz|2006|pp=291–292, 297–298, 302}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Viète and the Formal Equation, § Analytic Geometry}} }}</ref> Some historians see this development as a key turning point in the history of algebra and consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation.<ref>{{multiref | {{harvnb|Hazewinkel|1994|p=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73]}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} }}</ref>

In the 17th and 18th centuries, many attempts were made to find general solutions to polynomials of degree five and higher. All of them failed.<ref name="auto3">{{multiref | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Impasse with Radical Methods}} }}</ref> At the end of the 18th century, the German mathematician [[Carl Friedrich Gauss]] proved the [[fundamental theorem of algebra]], which describes the existence of [[Zero of a function|zeros]] of polynomials of any degree without providing a general solution.<ref name="auto7"/> At the beginning of the 19th century, the Italian mathematician [[Paolo Ruffini]] and the Norwegian mathematician [[Niels Henrik Abel]] were [[Abel–Ruffini theorem|able to show]] that no general solution exists for polynomials of degree five and higher.<ref name="auto3"/> In response to and shortly after their findings, the French mathematician [[Évariste Galois]] developed what came later to be known as [[Galois theory]], which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation of [[group theory]].<ref name="auto2"/> Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory.<ref>{{multiref | {{harvnb|Corry|2024|loc=§ Applications of Group Theory}} | {{harvnb|Bueno|French|2018|pp=[https://books.google.com/books?id=kmZaDwAAQBAJ&pg=PA73 73–75]}} }}</ref>

[[File:Garrett Birkhoff.jpeg|thumb|alt=Photo of Garrett Birkhoff|[[Garrett Birkhoff]] developed many of the foundational concepts of universal algebra.|left|upright=0.8]]

Starting in the mid-19th century, interest in algebra shifted from the study of polynomials associated with elementary algebra towards a more general inquiry into algebraic structures, marking the emergence of [[abstract algebra]]. This approach explored the axiomatic basis of arbitrary algebraic operations.<ref>{{multiref | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Corry|2024|loc=§ Structural Algebra}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73–74]}} }}</ref> The invention of new algebraic systems based on different operations and elements accompanied this development, such as [[Boolean algebra]], [[Vector space|vector algebra]], and [[matrix algebra]].<ref>{{multiref | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} | {{harvnb|Tanton|2005|p=10}} | {{harvnb|Corry|2024|loc=§ Matrices, § Quaternions and Vectors}} }}</ref> Influential early developments in abstract algebra were made by the German mathematicians [[David Hilbert]], [[Ernst Steinitz]], and [[Emmy Noether]] as well as the Austrian mathematician [[Emil Artin]]. They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, like groups, rings, and fields.<ref>{{multiref | {{harvnb|Merzlyakov|Shirshov|2020|loc=§ Historical Survey}} | {{harvnb|Corry|2024|loc=§ Hilbert and Steinitz, § Noether and Artin}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA73 73–74]}} }}</ref>

The idea of the even more general approach associated with universal algebra was conceived by the English mathematician [[Alfred North Whitehead]] in his 1898 book ''A Treatise on Universal Algebra''. Starting in the 1930s, the American mathematician [[Garrett Birkhoff]] expanded these ideas and developed many of the foundational concepts of this field.<ref>{{multiref | {{harvnb|Grätzer|2008|p=[https://books.google.com/books?id=8lNkXPJas4wC&pg=PR7 vii]}} | {{harvnb|Chang|Keisler|1990|p=[https://books.google.com/books?id=uiHq0EmaFp0C&pg=PA603 603]}} | {{harvnb|Knoebel|2011|p=[https://books.google.com/books?id=VWS_sgO2uvgC&pg=PA5 5]}} | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA74 74–75]}} }}</ref> The invention of universal algebra led to the emergence of various new areas focused on the algebraization of mathematics{{em dash}}that is, the application of algebraic methods to other branches of mathematics. Topological algebra arose in the early 20th century, studying algebraic structures such as [[topological groups]] and [[Lie groups]].<ref>{{multiref | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA74 74–75]}} | {{harvnb|Kleiner|2007|p=[https://books.google.com/books?id=udj-1UuaOiIC&pg=PA100 100]}} | {{harvnb|Carlson|2024|loc=§ History of topology}} }}</ref> In the 1940s and 50s, [[homological algebra]] emerged, employing algebraic techniques to study [[Homology (mathematics)|homology]].<ref>{{multiref | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA74 74–75]}} | {{harvnb|Weibel|1995|p=[https://books.google.com/books?id=UtIhAwAAQBAJ&pg=PR11 xi, 4]}} }}</ref> Around the same time, [[category theory]] was developed and has since played a key role in the [[foundations of mathematics]].<ref>{{multiref | {{harvnb|Krömer|2007|p=[https://books.google.com/books?id=41bHxtHxjUAC&pg=PA61 61]}} | {{harvnb|Laos|1998|p=[https://books.google.com/books?id=1r7dSn4ZqogC&pg=PA100 100]}} }}</ref> Other developments were the formulation of [[model theory]] and the study of [[free algebra]]s.<ref>{{multiref | {{harvnb|Hazewinkel|1994|pp=[https://books.google.com/books?id=PE1a-EIG22kC&pg=PA74 74–75]}} | {{harvnb|Pratt|2022|loc=§ 6. Free Algebras}} }}</ref>

{{clear}}

== Applications ==
{{see also|Applied mathematics}}
The influence of algebra is wide-reaching, both within mathematics and in its applications to other fields.<ref>{{multiref | {{harvnb|Houston|2004|p=[https://books.google.com/books?id=jsWL_XJt-dMC&pg=PA319 319]}} | {{harvnb|Neri|2019|p=[https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PR12 xii]}} | {{harvnb|Lidl|Pilz|1997|pp=[https://books.google.com/books?id=_49AmSWo4_AC&pg=PR7 vii–viii]}} }}</ref> The algebraization of mathematics is the process of applying algebraic methods and principles to other [[branches of mathematics]], such as [[geometry]], [[topology]], [[number theory]], and [[calculus]]. It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other.<ref>{{multiref | {{harvnb|Kleiner|2007|p=[https://books.google.com/books?id=udj-1UuaOiIC&pg=PA100 100]}} | {{harvnb|Pratt|2022|loc=§ 5. Algebraization of Mathematics}} | {{harvnb|Maddocks|2008|p=130}} | {{harvnb|Pratt|2022|loc=§ 5. Algebraization of Mathematics}} | {{harvnb|Mancosu|1999|pp=[https://books.google.com/books?id=60qaEePdqcoC&pg=PA84 84–85]}} }}</ref>

[[File:Sphere Quadric.png|thumb|alt=Rendered image of a sphere|The algebraic equation <math>x^2 + y^2 + z^2 = 1</math> describes a [[sphere]] at the [[Origin (mathematics)|origin]] with a radius of 1.]]

One application, found in geometry, is the use of algebraic statements to describe geometric figures. For example, the equation <math>y = 3x - 7</math> describes a line in two-dimensional space while the equation <math>x^2 + y^2 + z^2 = 1</math> corresponds to a [[sphere]] in three-dimensional space. Of special interest to [[algebraic geometry]] are [[algebraic varieties]],{{efn|Algebraic varieties studied in geometry differ from the more general varieties studied in universal algebra.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 1.4 Cartesian geometry, § 3. Universal Algebra}} | {{harvnb|Danilov|2006|p=[https://books.google.com/books?id=-QMWR-x66XUC&pg=PA174 174]}} }}</ref>}} which are solutions to [[systems of polynomial equations]] that can be used to describe more complex geometric figures.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 5.1 Algebraic Geometry}} | {{harvnb|Danilov|2006|pp=[https://books.google.com/books?id=-QMWR-x66XUC&pg=PA172 172, 174]}} }}</ref> Algebraic reasoning can also solve geometric problems. For example, one can determine whether and where the line described by <math>y = x + 1</math> intersects with the circle described by <math>x^2 + y^2 = 25</math> by solving the system of equations made up of these two equations.<ref>{{harvnb|Vince|2007|p=[https://books.google.com/books?id=B574tQbP6WcC&pg=PA133 133]}}</ref> Topology studies the properties of geometric figures or [[topological space]]s that are preserved under operations of [[continuous deformation]]. [[Algebraic topology]] relies on algebraic theories such as [[group theory]] to classify topological spaces. For example, [[homotopy groups]] classify topological spaces based on the existence of [[Loop (topology)|loops]] or [[Hole#In mathematics|holes]] in them.<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 5.3 Algebraic Topology}} | {{harvnb|Rabadan|Blumberg|2019|pp=[https://books.google.com/books?id=2967DwAAQBAJ&pg=PA49 49–50]}} | {{harvnb|Nakahara|2018|p=[https://books.google.com/books?id=p2C1DwAAQBAJ&pg=PA121 121]}} | {{harvnb|Weisstein|2003|pp=52–53}} }}</ref>

Number theory is concerned with the properties of and relations between integers. [[Algebraic number theory]] applies algebraic methods and principles to this field of inquiry. Examples are the use of algebraic expressions to describe general laws, like [[Fermat's Last Theorem]], and of algebraic structures to analyze the behavior of numbers, such as the [[ring of integers]].<ref>{{multiref | {{harvnb|Pratt|2022|loc=§ 5.2 Algebraic Number Theory}} | {{harvnb|Jarvis|2014|p=[https://books.google.com/books?id=0j0qBAAAQBAJ&pg=PA1 1]}} | {{harvnb|Viterbo|Hong|2011|p=[https://books.google.com/books?id=d89QRR24jbMC&pg=PA127 127]}} }}</ref> The related field of [[combinatorics]] uses algebraic techniques to solve problems related to counting, arrangement, and combination of discrete objects. An example in [[algebraic combinatorics]] is the application of group theory to analyze [[Graph (discrete mathematics)|graphs]] and symmetries.<ref>{{multiref | {{harvnb|Gowers|Barrow-Green|Leader|2010|pp=[https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA550 550, 561]}} | {{harvnb|Godsil|2017|p=[https://books.google.com/books?id=hLs6DwAAQBAJ&pg=PR8 viii]}} | {{harvnb|Betten|Kohnert|Laue|Wassermann|2013|p=[https://books.google.com/books?id=G1TmCAAAQBAJ&pg=PP9 ix]}} }}</ref> The insights of algebra are also relevant to calculus, which uses mathematical expressions to examine [[rates of change]] and [[Integral|accumulation]]. It relies on algebra, for instance, to understand how these expressions can be transformed and what role variables play in them.<ref>{{multiref | {{harvnb|Kilty|McAllister|2018|pp=x, 347, 589}} | {{harvnb|Bressoud|2021|p=[https://books.google.com/books?id=GkgHEAAAQBAJ&pg=PA64 64]}} }}</ref> [[Algebraic logic]] employs the methods of algebra to describe and analyze the structures and patterns that underlie [[logical reasoning]],<ref>{{multiref | {{harvnb|Halmos|1956|p=363}} | {{harvnb|Burris|Legris|2021|loc=§ 1. Introduction}} }}</ref> exploring both the relevant mathematical structures themselves and their application to concrete problems of logic.<ref>{{harvnb|Andréka|Németi|Sain|2001|pp=[https://link.springer.com/chapter/10.1007/978-94-017-0452-6_3 133–134]}}</ref> It includes the study of [[Boolean algebra]] to describe [[propositional logic]]<ref>{{multiref | {{harvnb|Andréka|Madarász|Németi|2020|loc=§ Concrete Algebraic Logic}} | {{harvnb|Pratt|2022|loc=§ 5.4 Algebraic Logic}} | {{harvnb|Plotkin|2012|pp=[https://books.google.com/books?id=-v3xCAAAQBAJ&pg=PA155 155–156]}} | {{harvnb|Jansana|2022|loc=Lead section}} }}</ref> as well as the formulation and analysis of algebraic structures corresponding to more complex [[Logic#Systems_of_logic|systems of logic]].<ref>{{multiref | {{harvnb|Andréka|Madarász|Németi|2020|loc=§ Abstract Algebraic Logic}} | {{harvnb|Jansana|2022|loc=§ 4. Algebras}} }}</ref>

[[File:Rubik's cube.svg|thumb|upright=0.8|alt=Picture of Rubik's cube|The faces of a [[Rubik's cube]] can be rotated to change the arrangement of colored patches. The resulting permutations form a group called the [[Rubik's Cube group]].<ref>{{harvnb|Joyner|2008|p=92}}</ref>]]

Algebraic methods are also commonly employed in other areas, like the natural sciences. For example, they are used to express [[scientific laws]] and solve equations in [[physics]], [[chemistry]], and [[biology]].<ref>{{multiref | {{harvnb|Houston|2004|p=[https://books.google.com/books?id=jsWL_XJt-dMC&pg=PA319 319]}} | {{harvnb|Neri|2019|p=[https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PR12 xii]}} | {{harvnb|Anton|Rorres|2010|p=[https://books.google.com/books?id=1PJ-WHepeBsC&pg=PA327 327]}} }}</ref> Similar applications are found in fields like [[economics]], [[geography]], [[engineering]] (including [[electronics]] and [[robotics]]), and [[computer science]] to express relationships, solve problems, and model systems.<ref>{{multiref | {{harvnb|Neri|2019|p=[https://books.google.com/books?id=NMOlDwAAQBAJ&pg=PR12 xii]}} | {{harvnb|Aleskerov|Ersel|Piontkovski|2011|pp=[https://books.google.com/books?id=ipcSD8ZGB8cC&pg=PA1 1–9]}} | {{harvnb|Straffin|1980|p=[https://www.jstor.org/stable/2689388 269]}} | {{harvnb|Menini|Oystaeyen|2017|p=[https://books.google.com/books?id=3mlQDwAAQBAJ&pg=PR5 v]}} | {{harvnb|Lovett|2015|p=[https://books.google.com/books?id=jRUqCgAAQBAJ&pg=PR9 ix]}} | {{harvnb|Lidl|Pilz|1997|pp=[https://books.google.com/books?id=_49AmSWo4_AC&pg=PR7 vii–viii]}} }}</ref> Linear algebra plays a central role in [[artificial intelligence]] and [[machine learning]], for instance, by enabling the efficient processing and analysis of large [[dataset]]s.<ref>{{multiref | {{harvnb|Wheeler|2023|p=[https://books.google.com/books?id=IqQIEQAAQBAJ&pg=PA29 29, 36–37]}} | {{harvnb|Gallier|Quaintance|2020|pp=[https://books.google.com/books?id=jWLcDwAAQBAJ&pg=PA1 1–2]}} }}</ref> Various fields rely on algebraic structures investigated by abstract algebra. For example, physical sciences like [[crystallography]] and [[quantum mechanics]] make extensive use of group theory,<ref>{{multiref | {{harvnb|Klimov|2014|p=[https://books.google.com/books?id=DZK3DwAAQBAJ&pg=PR9 ix]}} | {{harvnb|Bengtsson|Życzkowski|2017|pp=313–353}} }}</ref> which is also employed to study puzzles such as [[Sudoku]] and [[Rubik's cube]]s,<ref>{{harvnb|Terras|2019|pp=63–64, 142}}</ref> and [[Mathematics of paper folding|origami]].<ref>{{harvnb|Hull|2021|p=[https://books.google.com/books?id=LdX7DwAAQBAJ&pg=PA180 180]}}</ref> Both [[coding theory]] and [[cryptology]] rely on abstract algebra to solve problems associated with [[data transmission]], like avoiding the effects of [[noise]] and ensuring [[data security]].<ref>{{multiref | {{harvnb|Lidl|Pilz|1997|pp=183–184, 239–240}} | {{harvnb|Carstensen|Fine|Rosenberger|2011|pp=[https://books.google.com/books?id=Xo6iSxRfXz0C&pg=PA326 326–327]}} }}</ref>

== Education ==
{{See also|Mathematics education}}

[[File:Balance scale.svg|thumb|upright=1.3|class=skin-invert-image|alt=Diagram of a balance scale|[[Balance scales]] are used in algebra education to help students understand how equations can be transformed to determine unknown values.<ref>{{multiref | {{harvnb|Kieran|2006|p=[https://books.google.com/books?id=OTCsKu0BZ0kC&pg=PA15 15]}} | {{harvnb|Kaput|2018|p=[https://books.google.com/books?id=1GUPEAAAQBAJ&pg=PA186 186]}} |{{harvnb|Gardella|DeLucia|2020|pp=[https://books.google.com/books?id=HBXFDwAAQBAJ&pg=PA19 19–22]}} }}</ref>]]

Algebra education mostly focuses on elementary algebra, which is one of the reasons why elementary algebra is also called school algebra. It is usually not introduced until [[secondary education]] since it requires mastery of the fundamentals of arithmetic while posing new cognitive challenges associated with abstract reasoning and generalization.<ref>{{multiref | {{harvnb|Arcavi|Drijvers|Stacey|2016|p=xiii}} | {{harvnb|Dekker|Dolk|2011|p=[https://books.google.com/books?id=7sVFaMhwackC&pg=PA69 69]}} }}</ref> It aims to familiarize students with the formal side of mathematics by helping them understand mathematical symbolism, for example, how variables can be used to represent unknown quantities. An additional difficulty for students lies in the fact that, unlike arithmetic calculations, algebraic expressions are often difficult to solve directly. Instead, students need to learn how to transform them according to certain laws, often to determine an unknown quantity.<ref>{{multiref | {{harvnb|Arcavi|Drijvers|Stacey|2016|pp=2–5}} | {{harvnb|Drijvers|Goddijn|Kindt|2011|pp=[https://books.google.com/books?id=7sVFaMhwackC&pg=PA8 8–10, 16–18]}} }}</ref>

Some tools to introduce students to the abstract side of algebra rely on concrete models and visualizations of equations, including geometric analogies, manipulatives including sticks or cups, and "function machines" representing equations as [[flow diagram]]s. One method uses [[balance scales]] as a pictorial approach to help students grasp basic problems of algebra. The mass of some objects on the scale is unknown and represents variables. Solving an equation corresponds to adding and removing objects on both sides in such a way that the sides stay in balance until the only object remaining on one side is the object of unknown mass.<ref>{{multiref | {{harvnb|Kieran|2006|p=[https://books.google.com/books?id=OTCsKu0BZ0kC&pg=PA15 15]}} | {{harvnb|Kaput|2018|p=[https://books.google.com/books?id=1GUPEAAAQBAJ&pg=PA186 186]}} |{{harvnb|Gardella|DeLucia|2020|pp=[https://books.google.com/books?id=HBXFDwAAQBAJ&pg=PA19 19–22]}} | {{harvnb|Star|Foegen|Larson|McCallum|2015|pp=16–17}} }}</ref> [[Word problem (mathematics education)|Word problems]] are another tool to show how algebra is applied to real-life situations. For example, students may be presented with a situation in which Naomi's brother has twice as many apples as Naomi. Given that both together have twelve apples, students are then asked to find an algebraic equation that describes this situation (<math>2x + x = 12</math>) and to determine how many apples Naomi has {{nowrap|(<math>x = 4</math>).<ref>{{multiref | {{harvnb|Arcavi|Drijvers|Stacey|2016|pp=58–59}} | {{harvnb|Drijvers|Goddijn|Kindt|2011|p=[https://books.google.com/books?id=7sVFaMhwackC&pg=PA13 13]}} }}</ref>}}

At the university level, mathematics students encounter advanced algebra topics from linear and abstract algebra. Initial [[undergraduate]] courses in linear algebra focus on matrices, vector spaces, and linear maps. Upon completing them, students are usually introduced to abstract algebra, where they learn about algebraic structures like groups, rings, and fields, as well as the relations between them. The curriculum typically also covers specific instances of algebraic structures, such as the systems of rational numbers, the real numbers, and the polynomials.<ref>{{harvnb|Hausberger|Zandieh|Fleischmann|2021|pp=147–148}}</ref>

== See also ==
{{div col|colwidth=30em}}
* {{annotated link|Algebra over a set}}
* {{annotated link|Algebra tile}}
* {{annotated link|Algebraic combinatorics}}
* {{annotated link|C*-algebra}}
* {{annotated link|Clifford algebra}}
* {{annotated link|Commutative algebra}}
* {{annotated link|Composition algebra}}
* {{annotated link|Computer algebra}}
* {{annotated link|Cyclotomic polynomial}}
* {{annotated link|Diophantine equation}}
* {{annotated link|Discrete group}}
* {{annotated link|Dual space}}
* {{annotated link|Eigenvalues and eigenvectors}}
* {{annotated link|Equivalence class}}
* {{annotated link|Equivalence relation}}
* {{annotated link|Exterior algebra}}
* {{annotated link|F-algebra}}
* {{annotated link|Finite field}}
* {{annotated link|Fundamental theorem of finitely generated abelian groups}}
* {{annotated link|Geometric algebra}}
* {{annotated link|Heyting algebra}}
* {{annotated link|Hilbert space}}
* {{annotated link|Hilbert's Nullstellensatz}}
* {{annotated link|Hilbert's syzygy theorem}}
* {{annotated link|Hopf algebra}}
* {{annotated link|Lattice (group)}}
* {{annotated link|Lie group}}
* {{annotated link|Linear form}}
* {{annotated link|Linear subspace}}
* {{annotated link|Matrix decomposition}}
* {{annotated link|Multilinear map}}
* {{annotated link|Non-associative algebra}}
* {{annotated link|Outline of algebra}}
* {{annotated link|Quaternion}}
* {{annotated link|Rational function}}
* {{annotated link|Relational algebra}}
* {{annotated link|Representation theory}}
* {{annotated link|Root of unity}}
* {{annotated link|Scheme theory}}
* {{annotated link|Sigma-algebra}}
* {{annotated link|Singular value decomposition}}
* {{annotated link|Spectral theory}}
* {{annotated link|Symmetric algebra}}
* {{annotated link|T-algebra}}
* {{annotated link|Tensor}}
* {{annotated link|Tensor algebra}}
* {{annotated link|Wiles' proof of Fermat's Last Theorem}}
{{div col end}}

== References ==
=== Notes ===
{{notelist}}

=== Citations ===
{{Reflist}}

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* {{cite book |last1=Waerden |first1=Bartel L. van der |title=A History of Algebra: From al-Khwārizmī to Emmy Noether |date=2013 |publisher=Springer |isbn=978-3-642-51599-6 |url=https://books.google.com/books?id=W6DwCAAAQBAJ |language=en |access-date=January 27, 2024 }}{{cbignore}}
* {{cite book |last1=Wagner |first1=Sigrid |last2=Kieran |first2=Carolyn |title=Research Issues in the Learning and Teaching of Algebra: The Research Agenda for Mathematics Education |volume=4 |date=2018 |publisher=Routledge |isbn=978-1-135-43421-2 |url=https://books.google.com/books?id=uW4ECwAAQBAJ&pg=PT225 |language=en |access-date=January 13, 2024 }}{{cbignore}}
* {{cite book |last1=Walz |first1=Guido |title=Lexikon der Mathematik: Band 1: A bis Eif |date=2016 |publisher=Springer |isbn=978-3-662-53498-4 |url=https://www.spektrum.de/lexikon/mathematik/algebra/12062 |language=de |chapter=Algebra |access-date=January 13, 2024 |archive-date=January 12, 2024 |archive-url=https://web.archive.org/web/20240112171819/https://www.spektrum.de/lexikon/mathematik/algebra/12062 |url-status=live |trans-title=Encyclopedia of Mathematics: Volume 1: A to Eif }}
* {{cite book |last1=Weibel |first1=Charles A. |title=An Introduction to Homological Algebra |publisher=Cambridge University Press |isbn=978-1-139-64307-8 |url=https://books.google.com/books?id=UtIhAwAAQBAJ&pg=PR11 |language=en |date=1995 }}
* {{cite book |last1=Weisstein |first1=Eric W. |title=CRC Concise Encyclopedia of Mathematics |date=2003 |publisher=Chapman & Hall/CRC |isbn=978-1-58488-347-0 |edition=2nd }}
* {{cite book |last1=Wheeler |first1=Jeffrey Paul |title=An Introduction to Optimization with Applications in Machine Learning and Data Analytics |publisher=CRC Press |isbn=978-1-003-80359-1 |url=https://books.google.com/books?id=IqQIEQAAQBAJ&pg=PA29 |language=en |date=2023 }}
* {{cite book |last1=Whitelaw |first1=T. A. |title=Introduction to Abstract Algebra, Third Edition |publisher=CRC Press |isbn=978-0-7514-0147-9 |url=https://books.google.com/books?id=f2hyf0QoB_0C&pg=PA61 |language=en |date=1995 }}{{cbignore}}
* {{cite book |last1=Williams |first1=G. Arnell |title=Algebra the Beautiful: An Ode to Math's Least-Loved Subject |date=2022 |publisher=Basic Books |isbn=978-1-5416-0070-6 |url=https://books.google.com/books?id=581CEAAAQBAJ&pg=PT62 |language=en |access-date=March 11, 2024 }}{{cbignore}}
* {{cite book | last=Wilson | first=Robert A. | author-link = Robert Arnott Wilson | title=The Finite Simple Groups | publisher=[[Springer-Verlag]]  | series=[[Graduate Texts in Mathematics]] | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | year=2009 | zbl=1203.20012 | volume=251 }}
* {{cite book |last1=Young |first1=Cynthia Y. |title=Precalculus |date=2010 |publisher=John Wiley & Sons |isbn=978-0-471-75684-2 |url=https://books.google.com/books?id=9HRLAn326zEC&pg=RA1-PA999 |language=en |access-date=January 16, 2024 }}{{cbignore}}
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* {{cite web |author1=zbMATH Open |title=Classification |url=https://zbmath.org/classification/ |website=zbMATH Open |publisher=Mathematical Reviews and zbMATH Open |access-date=March 17, 2024 |language=en |date=2024 |archive-date=July 19, 2020 |archive-url=https://web.archive.org/web/20200719184539/https://zbmath.org/classification/ |url-status=live }}
* {{cite book |last1=Zwillinger |first1=Daniel |title=CRC Standard Mathematical Tables and Formulae |date=2002 |publisher=CRC Press |isbn=978-1-4200-3534-6 |url=https://books.google.com/books?id=gE_MBQAAQBAJ&pg=PA812 |language=en |access-date=January 27, 2024 }}{{cbignore}}
{{Refend}}

== External links ==
{{Wikiquote}}
{{Wiktionary|algebra}}
{{Wikibooks|Algebra}}
{{EB1911 poster|Algebra}}

{{Algebra |expanded}}
{{Polynomials}}
{{Areas of mathematics |collapsed}}
{{Authority control}}

[[Category:Algebra| ]]