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{{Short description|Solution in radicals of a polynomial equation}}
{{Confuse|Algebraic number}}

An '''algebraic solution''' or '''solution in radicals''' is a [[Closed-form expression|closed-form expression]], and more specifically a closed-form [[Algebraic expression|algebraic expression]], that is the solution of an [[Algebraic equation|algebraic equation]] in terms of the coefficients, relying only on [[Addition|addition]], [[Subtraction|subtraction]], [[Multiplication|multiplication]], [[Division (mathematics)|division]], raising to integer powers, and the extraction of [[Nth root|nth root]]s (square roots, cube roots, and other integer roots). 

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

There exist more complicated algebraic solutions for [[Cubic equation|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|quartic equation]]s.<ref>Carpenter, William, "On the solution of the real quartic," ''Mathematics Magazine'' 39, 1966, 28-30.</ref>  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 equations, 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|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|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.

[[Biography:Évariste Galois|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.

Algebraic solutions form a subset of [[Closed-form expression|closed-form expression]]s, because the latter permit [[Transcendental function|transcendental function]]s (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.

==See also==
*Solvable quintics
*[[Sextic equation#Solvable sextics|Solvable sextics]]
*[[Septic equation#Solvable septics|Solvable septics]]

==References==
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{{DEFAULTSORT:Algebraic Solution}}
[[Category:Algebra]]
[[Category:Equations]]


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