Algebraic solution

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 = − b ± b 2 − 4 a c   2 a , {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}},}

which expresses the solutions of the quadratic equation

a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.}

There exist algebraic solutions for cubic equations^[1] and quartic equations,^[2] which are more complicated than the quadratic formula. The Abel–Ruffini theorem,^[3]: 211 and, more generally Galois theory, state that some quintic equations, such as

x 5 − x + 1 = 0 , {\displaystyle 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 {\displaystyle x^{10}=2} can be solved as x = ± 2 10 . {\displaystyle x=\pm {\sqrt[{10}]{2}}.} The eight other solutions are nonreal complex numbers, which are also algebraic and have the form x = ± r 2 10 , {\displaystyle 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.

See also

• Radical symbol
• Solvable quintics
• Solvable sextics
• Solvable septics

References