Is Galois' groundbreaking paper on solvable groups still relevant today?

[jostpuur]

When a Lie algebra is solvable, does it have something to do with actually solving something?

 

[HallsofIvy]

A group, G, (whether "Lie" or not) is said to be solvable if and only if there exist a sequence of subgroups, G₁, G₂, ..., G_n of G, such that:
G₁ is the subgroup consisting only of the group identity,
G_n is G itself,
Each G_i is a normal subgroup of G_i+1, and
G_i+1/G_i is commutative.

The name, and indeed the whole definition, is from Galois's answer to the question of whether or not there could exist a "formula" for polynomials of degree 5 or higher, in terms of radicals.

The point is that a polynomial equation, with integer coefficients, is "solvable by radicals" if and only if its Galois group is a "solvable" group.
I think (I'm entering unsure ground here) that Galois also showed that, for n> 4, there exist a polynomial of degree n whose Galois group is all of S_n (the group of permutations on n objects) and that S_n, for n> 4, is not a "solvable" group.

 

[Chris Hillman]

Quick Clarification

In his 1831 paper, Galois (age 20) defined a group for any polynomial with rational coefficients, a permutation group acting on the roots, which we now call the galois group of the polynomial. He showed (in essence) that this group acts transitively iff the polynomial is irreducible, and if so, it is solvable iff the roots of the polynomial can be expressed in closed form in terms of arithmetic operations plus the extraction of roots (i.e. the problem of finding the roots can be reduced to finding the roots of monomials). In such cases, the roots are expressed as nested radicals, in a manner which mirrors the descending subnormal chain of subgroups (the composition series exhibiting the fact that the group is indeed solvable--- the composition series can be refined so that the quotients mentioned by Halls are in fact prime order cyclic groups, corresponding to monomials of prime degree). In modern terms, he exhibited a galois duality between the subgroups of the galois group and the intermediate extension fields of the extension field of the rationals defined by adjoining the roots of our polynomial to the field of rational numbers. Galois also showed that while "special" polynomials of high degree can have solvable galois groups, the group of a "generic" n-degree irreducible polynomial is the symmetric group on n letters, $S_n$, which is not solvable for n > 4 (indeed its index two normal subgroup $A_n$ is not solvable for n > 4). Thus, there can be no general formula analogous to the quadratic formula for the quintic or higher degrees.

In order to do all this, since groups, rings, and fields (and group actions, and finite projective planes...) did not yet exist, he had to invent them. Indeed, most would agree that he invented modern algebra. All this in a dozen pages, yet his paper clearly contains all the essential ideas, albeit in sketchy and sometimes delphic form.

This paper is often regarded as one of the single most profound advances in mathematical thought, because Galois was the first to clearly see that algebraic objects more complicated than numbers are worthy of recognition and study. In a sense, he took the first step down the path of categorification. As we recently discussed in some other PF threads, he also introduced one of the great themes of mathematics, the notion of symmetry (and its relation to the notion of information).

His remarkable achievement in completely resolving a problem which had remained unsolved for millenia, in an utterly original and completely unexpected manner, inspired many of the greatest nineteenth century mathematicians, including Sophus Lie (whose dream of doing for differential equations what Galois had done for polynomials led to the development of Lie theory as required background for Lie's theory of the symmetries of differential equations). And Galois continues to inspire mathematicians today (one might mention Grothendieck as a more recent example).

For the simplest example of nested radicals, see Cardano's formula for the roots of a cubic:
http://gowers.wordpress.com/2007/09/15/discovering-a-formula-for-the-cubic/