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 which expresses the solutions of the quadratic equation 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 do not have any algebraic solution.
The eight other solutions are nonreal complex numbers, which are also algebraic and have the form
Évariste Galois introduced a criterion allowing one to decide which equations are solvable in radicals.