Abel–Ruffini theorem

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In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.[1]

Contents

Interpretation

The content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are unsolvable. In fact, the opposite is true: every polynomial equation in one unknown, with real or complex coefficients, has at least one complex number as solution; this is the fundamental theorem of algebra. Although the solutions cannot always be expressed exactly with radicals, 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 theorem says that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational constants, and repeatedly forming sums, differences, products, quotients, and radicals (n-th roots, for some integer n) of previously obtained numbers. This clearly excludes the possibility of having any formula that expresses the solutions of an arbitrary equation of degree 5 or higher in terms of its coefficients, using only those operations, or even of having different formulas for different roots or for different classes of polynomials, in such a way as to cover all cases. (In principle one could imagine formulas using irrational numbers as constants, but even if a finite number of those were admitted at the start, not all roots of higher-degree equations could be obtained.) However some polynomial equations, of arbitrarily high degree, are solvable with such operations. Indeed if the roots happen to be rational numbers, they can trivially be expressed as constants. The simplest nontrivial example is the monomial equation axn = b, whose solutions are

Here the expression eik / n, which appears to involve the use of the exponential function, in fact just gives different the possible values of $\sqrt[n]1$ (the n-th roots of unity), so it involves only extraction of radicals.