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
of the quadratic equation
There exist more complicated algebraic solutions for cubic equations and quartic equations. The Abel–Ruffini theorem,: 211 and, more generally Galois theory, state that some quintic equations, such as
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.
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.
- 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