Solution in radicals

Short description: Solution in radicals of a polynomial equation

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

[math]\displaystyle{ x=\frac{-b \pm \sqrt {b^2-4ac\ }}{2a} }[/math]

of the quadratic equation

[math]\displaystyle{ ax^2 + bx + c =0. }[/math]

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

[math]\displaystyle{ 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]\displaystyle{ x^{10} = 2 }[/math] can be solved as [math]\displaystyle{ x=\pm\sqrt[10]2. }[/math] The eight other solutions are nonreal complex numbers, which are also algebraic and have the form [math]\displaystyle{ x=\pm r\sqrt[10]2, }[/math] 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