solution in radicals

{{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 of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of nth roots (square roots, cube roots, etc.).

A well-known example is the quadratic formula

:$x=\backslash frac\{-b\; \backslash pm\; \backslash sqrt\; \{b^2-4ac\backslash \; \}\}\{2a\},$

which expresses the solutions of the quadratic equation

:$ax^2\; +\; bx\; +\; c\; =0.$

There exist algebraic solutions for cubic equationsNickalls, 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. and quartic equations,Carpenter, William, "On the solution of the real quartic," Mathematics Magazine 39, 1966, 28-30. which are more complicated than the quadratic formula. The Abel–Ruffini theorem,Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, {{ISBN|978-0-486-47189-1}}{{rp|211}} and, more generally Galois theory, state that some quintic equations, such as

:$x^5-x+1=0,$

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 $x^\{10\}\; =\; 2$ can be solved as $x=\backslash pm\backslash sqrt[10]2.$ The eight other solutions are nonreal complex numbers, which are also algebraic and have the form $x=\backslash pm\; r\backslash sqrt[10]2,$ where {{mvar|r}} is a fifth root of unity, which can be expressed with two 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.