Galois Theory - History

History

See also: Abstract algebra#Early group theory

Galois theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, (x – a)(x – b) = x2 – (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.

This was first formalized by the 16th century French mathematician François Viète, in Viète's formulas, for the case of positive real roots. In the opinion of the 18th century British mathematician Charles Hutton, the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th century French mathematician Albert Girard; Hutton writes:

... the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.

In this vein, the discriminant is a symmetric function in the roots which reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots. See Discriminant:Nature of the roots for details.

The cubic was first partly solved by the 15th/16th century Italian mathematician Scipione del Ferro, who did not however publish his results; this method only solved one of three classes, as the others involved taking square roots of negative numbers, and complex numbers were not known at the time. This solution was then rediscovered independently in 1535 by Niccolò Fontana Tartaglia, who shared it with Gerolamo Cardano, asking him to not publish it. Cardano then extended this to the other two cases, using square roots of negatives as intermediate steps; see details at Cardano's method. After the discovery of Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his complete solution in his 1545 Ars Magna. His student Lodovico Ferrari solved the quartic polynomial, which solution Cardano also included in Ars Magna.

A further step was the 1770 paper Réflexions sur la résolution algébrique des équations by the French-Italian mathematician Joseph Louis Lagrange, in his method of Lagrange resolvents, where he analyzed Cardano and Ferrarri's solution of cubics and quartics by considering them in terms of permutations of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois theory. Crucially, however, he did not consider composition of permutations. Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree.

The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation. His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel–Ruffini theorem.

While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved, such as (x − 1)5=0, and the precise criterion by which a given quintic or higher polynomial could be determined to be solvable or not was given by Évariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its Galois group – had a certain structure – in modern terms, whether or not it was a solvable group. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degree.