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{{Short description|Solution in radicals of a polynomial equation}}
{{Distinguish|Algebraic number}}
{{use mdy dates|date=September 2021}}
{{Use American English|date = January 2019}}

A '''solution in radicals''' or '''algebraic solution''' is an [[expression (mathematics)|expression]] of a solution of a [[polynomial equation]] that is [[algebraic expression|algebraic]], that is, relies only on [[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]], [[exponentiation|raising to integer powers]], and extraction of [[nth root|{{mvar|n}}th root]]s ([[square roots]], [[cube roots]], etc.).

A well-known example is the [[quadratic formula]] 
:<math>x=\frac{-b \pm \sqrt {b^2-4ac\  }}{2a},</math>
which expresses the solutions of the [[quadratic equation]]
:<math>ax^2 + bx + c =0.</math>

There exist algebraic solutions for [[cubic equation]]s<ref>Nickalls, R. W. D., "[http://img2.timg.co.il/forums/1_90809354.pdf A new approach to solving the cubic: Cardano's solution revealed]," ''Mathematical Gazette'' 77, November 1993, 354-359.</ref> and [[quartic equation]]s,<ref>Carpenter, William, "On the solution of the real quartic," ''Mathematics Magazine'' 39, 1966, 28-30.</ref> which are more complicated than the quadratic formula. The [[Abel–Ruffini theorem]],<ref>Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, {{ISBN|978-0-486-47189-1}}</ref>{{rp|211}} and, more generally [[Galois theory]], state that some [[quintic equation]]s, such as 
:<math>x^5-x+1=0,</math>
do not have any algebraic solution. The same is true for every higher degree. However, for any degree there are some polynomial equations that have algebraic solutions; for example, the equation <math>x^{10} = 2</math> can be solved as <math>x=\pm\sqrt[10]2.</math> The eight other solutions are nonreal [[complex number]]s, which are also algebraic and have the form <math>x=\pm r\sqrt[10]2,</math> where {{mvar|r}} is a fifth [[root of unity]], which can be expressed with two [[nested radical|nested square roots]]. See also {{slink|Quintic function|Other solvable quintics}} for various other examples in degree 5.

[[Évariste Galois]] introduced a criterion allowing one to decide which equations are solvable in radicals. See [[Radical extension]] for the precise formulation of his result.

==See also==
*[[Radical symbol]]
*[[quintic equation#Solvable quintics|Solvable quintics]]
*[[sextic equation#Solvable sextics|Solvable sextics]]
*[[septic equation#Solvable septics|Solvable septics]]

==References==
{{reflist}}

[[Category:Algebra]]
[[Category:Equations]]


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