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...ugel.svg|thumb|right|The [[Riemann sphere]] is one of the simplest complex algebraic varieties.]] ...Shafarevich]], eds. ''Algebraic Geometry III: Complex Algebraic Varieties. Algebraic Curves and Their Jacobians.'' Vol. 3. Springer, 1998. {{ISBN|3-540-54681-2} ...

...''K'' of the rational numbers '''Q''', such that the [[discriminant of an algebraic number field|discriminant]] of ''K''/'''Q''' is at most ''N''. The theorem {{cite book|last=Neukirch|first=Jürgen|title=Algebraic Number Theory|year=1999|publisher=Springer}} Section III.2 ...

{{About||the algebraic closure of a field|Algebraic closure}} '''Algebraic closure''' of a subset <math>A</math> of a vector space <math>X</math> is t ...

...and the coefficients of their [[power series]] expansion lie in a fixed [[algebraic number field]] and have heights of at most exponential growth. where the coefficients ''a_n'' all belong to the same [[algebraic number field]], ''K'', and with the following two properties. ...

...in [[real algebraic geometry]] that the [[radical of an ideal]] plays in [[algebraic geometry]] over an [[algebraically closed field]]. ...f ''I''. In real algebraic geometry, the Nullstellensatz fails as the real numbers are not algebraically closed. However, one can recover a similar theorem, t ...

{{Short description|Type of algebraic surface}} ...er words the Kodaira theorem, valid in algebraic geometry over the complex numbers, has such surfaces as counterexamples, and these can only exist in [[charac ...

...mr=997310}}</ref> It marked a breakthrough in the theory of transcendental numbers. Many longstanding open problems can be deduced as direct consequences. ...less <math>A</math> contains a proper [[Algebraic group#Algebraic subgroup|algebraic subgroup]]. ...

{{Short description|Special numbers in mathematics}} ...tor2-first=Hiroaki | editor3-last=Takahashi | editor3-first=Hiroki | title=Algebraic number theory and related topics 2007 | publisher=[[Research Institute for ...

...to be strictly positive (so that division by zero and other inappropriate algebraic operations are not encountered). ...efficients <math>c_i</math> and the exponents <math>a_{ij}</math> are real numbers. Signomials are [[Closure (mathematics)|closed]] under addition, subtracti ...

In [[mathematics]], a '''monogenic field''' is an [[algebraic number field]] ''K'' for which there exists an element ''a'' such that the In a monogenic field ''K'', the [[Discriminant of an algebraic number field|field discriminant]] of ''K'' is equal to the [[discriminant]] ...

== Algebraic expressions == [[File:Exp-tree-ex-12.svg|thumb|270px|right|Binary algebraic expression tree equivalent to ((5 + z) / -8) * (4 ^ 2)]] ...

{{Short description|Identity for real numbers}} ...|Geometric interpretation of the Candido identity for sequential Fibonacci numbers. The white area equals the grey area and each of them equals half of the ou ...

...topology]]. A '''weak character''' is a restriction to ''&Lambda;'' of an algebraic homomorphism from ''&Lambda;''_''d'' into the [[circle group]]: Interesting examples of multiplicatively closed harmonious sets of real numbers arise in the theory of [[diophantine approximation]]. ...

...atics, especially in [[algebraic K-theory|algebraic ''K''-theory]], the '''algebraic ''K''-group of a field''' is important to compute. For a finite field, the ...computations of K-theory of global fields (such as [[number field]]s and [[Algebraic function field|function field]]s), as well as local fields (such as [[p-adi ...

...ral numbers. Natural numbers including 0 are also sometimes called ''whole numbers''.<ref> ...ath>\mathbb{Z}</math>): Positive and [[negative number|negative]] counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}. ...

...gorithm is therefore fundamental, and it is widely used in [[computational algebraic geometry]]. ==Failure with algebraic sets== ...

...generalises the idea of exponential functions on the ordered field of real numbers. ..., ''Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function'', J. Amer. M ...

...derburg]], [[Denmark]]) is a German mathematician, known for his work in [[algebraic geometry]]. ...tion]] for the [[Betti numbers]] of the [[Hilbert scheme]] of points on an algebraic [[Surface (topology)|surface]]: ...

In [[algebraic topology]], a '''commutative ring spectrum''', roughly equivalent to a [[E- ...ommutative ring spectra over the field <math>\mathbb{Q}</math> of rational numbers is [[Quillen equivalent]] to the category of [[differential graded algebra] ...

[[Image:Floor function.svg|thumb|right|288px|The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circl Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the [[Dirichlet function]], the [[ ...