Abel-Ruffini theorem

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The Abel-Ruffini theorem, often known just as Abel's theorem, states that it is impossible to solve a general quintic equation "in radicals". This is in stark contrast with polynomials of smaller degree.

Recall that a "quadratic polynomial" is an expression of the form p(x) = ax2 + bx + c, where a, b, and c are some real (or complex) coefficients. It is a familiar fact that the solutions of ax2 + bx + c are given by the quadratic formula as

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.

It was long wondered whether it was possible to give a similar formula for polynomials of larger degree, for example cubics ax3 + bx2 + cx + d = 0 and quartics ax4 + bx3 + cx2 + dx + e = 0. The explicit, but more elaborate, "cubic formula" and "quartic formula" were derived by Cardano and Ferrari by the 16th century. The first of these involves taking a number of cube roots, and the later a number of fourth roots. It was long wondered whether a similar formula existed for quintic polynomials, so that solutions could be computed through some complex formula involving fifth roots, but Abel's theorem states that this is impossible: there is no way to solve a general quintic "in radicals". A standard example of a quintic whose roots may not be expressed as radicals is x5x + 1 = 0.

The Abel-Ruffini theorem provided the impetus for the development of the modern field of Galois theory, and indeed much of abstract algebra. Once some amount of Galois theory is understood, the impossibility of solving a general quintic turns out to be a consequence of some easy facts in group theory: in particular, the fact that the permutation group S5 is not "solvable".

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