Solution in radicals

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 $n$th 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 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.