The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher.

The content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are insoluble. In fact, all these polynomial equations have solutions; this is the fundamental theorem of algebra. Although these solutions cannot always be computed exactly, they can be computed to any desired degree of accuracy using numerical methods such as the Newton-Raphson method or Laguerre method, and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees.

The theorem only concerns the form that such a solution must take. The content of the theorem is that the solution of a higher-degree equation cannot always be expressed by starting with the coefficients and using only the operations of addition, subtraction, multiplication, division and extracting roots (radicals).

For example, the solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction, multiplication, division, and square roots, using the familiar quadratic equation: The roots of ax² + bx + c = 0 are

Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, had been known since the 16th century.

The Abel-Ruffini theorem says that there are some fifth-degree equations whose solution cannot be so expressed. The equation x⁵ - x + 1 = 0 is an example. Some other fifth degree equations can be solved by radicals, for example x⁵ - x⁴ - x + 1 = 0. The precise criterion that distinguishes between those equations that can be solved by radicals and those that cannot was given by Evariste Galois and is now part of Galois theory: a polynomial equation can be solved by radicals if and only if its Galois group is a solvable group. In the modern analysis, the reason that second, third and fourth degree polynomial equations can always be solved by radicals while higher degree equations cannot is nothing but the algebraic fact that the symmetric groups S₂, S₃ and S₄ are solvable groups, while S_n is not solvable for n≥5.

History

The theorem was first proved by Paolo Ruffini in 1799, but his proof was mostly ignored. While it contained a minor gap, it was quite innovative in using permutation groups. The theorem is also credited to Niels Henrik Abel, who published a proof in 1824.