Solution in radicals of a polynomial equation
A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).
A well-known example is the solution
![{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb26d837baae040fcb46efebc4d1cdc8f721277)
of the quadratic equation
![{\displaystyle ax^{2}+bx+c=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70a0e43dfc81e6fea3be4fc96895a8f9ec2966ac)
There exist more complicated algebraic solutions for cubic equations[1] and quartic equations.[2] The Abel–Ruffini theorem,[3]: 211 and, more generally Galois theory, state that some quintic equations, such as
![{\displaystyle x^{5}-x+1=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f385b14b324e6dd45f9276db839ad740bb370491)
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
can be solved as
The eight other solutions are nonreal complex numbers, which are also algebraic and have the form
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.
Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.
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