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=\frac{-b \pm \sqrt {b^2-4ac\ }}{2a},
which expresses the solutions of the quadratic equation
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,
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 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
References
- ^ Nickalls, R. W. D., "A new approach to solving the cubic: Cardano's solution revealed," Mathematical Gazette 77, November 1993, 354-359.
- ^ Carpenter, William, "On the solution of the real quartic," Mathematics Magazine 39, 1966, 28-30.
- ^ Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1