Solution in radicals

Not to be confused with Algebraic number.

A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).

A well-known example is the solution

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

of the quadratic equation

$ax^{2}+bx+c=0.$

There exist more complicated algebraic solutions for cubic equations and quartic equations. The Abel–Ruffini theorem, 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 $x=\pm {\sqrt[{10}]{2}}.$ The eight other solutions are nonreal complex numbers, which are also algebraic and have the form $x=\pm r{\sqrt[{10}]{2}},$ where r is a fifth root of unity, which can be expressed with two nested square roots. See also 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.

Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.

See also

• Solvable quintics
• Solvable sextics
• Solvable septics