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 th roots ( square roots, cube roots, etc.).

A well-known example is the quadratic formula
:$x=\frac,$
which expresses the solutions of the quadratic equation
:$ax^2 + bx + c =0.$

There exist algebraic solutions for cubic equations and quartic equations, which are more complicated than the quadratic formula. The Abel–Ruffini theorem,Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, 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 $x^ = 2$ can be solved as x=\pm\sqrt 0. The eight other solutions are nonreal complex numbers, which are also algebraic and have the form x=\pm r\sqrt 0, where 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.

See also

* Radical symbol
* Solvable quintics
* Solvable sextics
* Solvable septics

References

Algebra
Equations

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