Solution in radicals

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 root|Template:Mvarth root]]s (square roots, cube roots, etc.).

A well-known example is the quadratic formula

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

which expresses the solutions of the quadratic equation

$a{x}^2+bx+c=0.$

There exist algebraic solutions for cubic equations and quartic equations, which are more complicated than the quadratic formula. The Abel–Ruffini theorem,Template:Rp 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 Template:Mvar is a fifth root of unity, which can be expressed with two nested square roots. See also Template:Slink 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

Template:Algebra-stub