Galois: what did he do and why do we still care?
phoenixthoth said:
What is Galois for other than determining solvability by radicals to certain polynomial equations?
I'll interpret that to mean: "what ideas did Galois introduce during his brief career in mathematics and what are they good for?"
That's a good question and I reget that I don't have time to offer a truly appropriate answer.
Just a few thoughts off the top of my head:
Galois more or less introduced groups, lattices (in sense of Mac Lane), rings, ideals and fields, which form the core of modern algebra. Along the way he more or less introduced the concept of object, morphism, and subobject, although these were not formalized until the last century with the introduction of category theory. I probably don't need to try to explain why modern algebra and category theory are important in modern mathematics!
In the nineteenth century, "geometry" pretty much meant "algebraic geometry", the geometry of curves and surfaces defined by sets of polynomial equations, especially in the setting of finite dimensional projective spaces over the complex field. Galois more or less introduced something completely novel, finite fields and projective planes over such fields, which are "geometries" with finitely many "points" and "lines". These turn out to have unexpected applications in such areas as algebraic coding theory, but the importance of his innovation goes far beyond finite projective planes themeselves. Indeed, the importance of finite projective planes wasn't recognized until the middle of the last century, when Fano rediscovered these geometries.
More generally, the idea of finding "geometries" hiding in "combinatorial relations" respecting some group action is essentially one half of Klein's Erlangen program, which offers one of the most influential (if least known) organizing principles in all of mathematics. To repeat what I said last time: every "geometrical configuration" has an algebraic doppelganger,an
invariant (perhaps an invariant polynomial or an invariant relation) of a suitable group action, and conversely every invariant under a group action (in fact, every
groupoid) expresses in algebraic terms some "geometrical relationship" among appropriate "geometric elements" (such as points, lines, and planes).
Turning to Galois theory specifically, as you no doubt know, in his famous 1831 paper, Galois completely solving the ancient problem of expressing the roots of polynomials in terms of radicals (or IOW reducing the problem of expressing the roots of a general polynomial to the roots of a monomial, which we assume we know how to accomplish--- as you probably know, Galois showed this can be done exactly in case the Galois group of the polynomial is
solvable. No mean achievement, but
sans doubt the most important aspect of this work was that Galois provided an inspiring example of one of the greatest ideas of mathematics: transform a mathematical problem expressed in terms of one theory into the language of another theory, solve it there, and transform the answer back into the original language. This is one of the truly great ideas in intellectual history (around the same time, Gauss was using it in what we now call differential geometry, but this wasn't published until around 1850).
Specifically, Galois more or less introduced what is now called a
Galois connection in lattice theory, an "order reversing bijection" between the lattice of "intermediate fields" of a field extension (a bit of anachronism--- these concepts are due to Artin and Noether around 1900) and the lattice of subgroups of the Galois group. The latter are much easier to find, exploiting the poweful and convenient theory of finite groups! Actually, this is the "almost trivial" part of Galois theory, but never underestimate a triviality; it turns out that Galois connections are everywhere in mathematics (and computer science). To mention just one, there is a Galois duality between algebraic varieties (curves and surfaces defined by sets of polynomial equations, the fundamental object of study in "classical" algebraic geometry) and ideals of a polynomial ring (the fundamental object of study in "classical" commutative algebra). See again mathwonk's posts in this thread for many references to this setting!
Well, I should mention one other example of a Galois duality: the one introduced by Galois himself, in which given any action by a group G on a set X, the lattice of pointwise stabilizer subgroups is dual to the lattice of fixsets. Here, the fixset of a subgroup H < G is the set of points, \triangleright H which is fixed by every element of the subgroup H; the stabilizer of a subset A of X is the set of elements of G, \triangleleft A, which fix every point in A. The fixset of the stabilizer of A, or \triangleright \triangleleft A, is a subset including A, but often much larger, sometimes called the Galois closure of A--- not to be confused with similar terminology in field theory! This can be characterized as the smallest set B such that given knowledge of where an unknown element g of G moves the points of A, you can deduce how g moves all the points of B. The homogeneous spaces G/\triangleleft A, where \triangleleft A is the stabilizer of A, were called
complexions by Planck and Boltzmann. Their "dimensions" (more or less) correspond to the "entropies" studied in information theory. (The latter subject, which soon grew into one of the most successful mathematical theories of all time, was only introduced by Shannon in 1948, but it could have been introduced much earlier, all the tools being available by 1870 at the latest!)
The "easy half" of Galois's theorem says that the problem of expressing the roots in terms of radicals is solvable only if the Galois group is solvable (has a
composition series, a chain of subgroups each normal in its parent supergroup). With some changes in language, this is true in a much more general context than the theory of fields. The "nontrivial half", the part essentially exploiting the theory of field extensions, is the converse: if the Galois group has a composition series, one can write the roots in terms of nested radicals, with the nesting and the degrees of the radicals corresponding to the
factor groups of the composition series, certain prime cyclic groups. (I am oversimplifying a bit.)
Even today, few books on Galois theory attempt to explain how this works--- I forgot to say in my previous post that while the book by Gaal doesn't discuss how to compute the Galois groups of specific polynomials, it does discuss the nontrivial half of Galois's paper, which outlined a construction. (This paper is translated in Edwards, Galois Theory, and as you will see it is quite sketchy--- nonetheless, everyone agrees that all the basic ideas which are required are presented there in embryonic form.) I'll just add that to prove the hard half, Galois brilliantly exploited one of Gauss's greatest innovations, modular arithmetic. Between them, Gauss and Galois are responsible for the fact that so many algebra books contain frequent repetitions of the phrase "reduce modulo a prime".
Last but not least, in his very first paper, Galois (IIRC he was 17 at the time) proved something fundamental about continued fractions. See for example the books by Khintchin or Olds. BTW, simple continued fractions turn out to be closely related to several things I hinted at: SL(2,C) is a basic example in theories of representations, invariants, reflection groups, and Lie theory, and the simple continued fraction algorithm is closely related to the euclidean algorithm, which is one of the fundamental
"algebraico-geometrical" algorithsm which is generalized by reduction modulo a Groebner basis. The latter technique (introduced as recently as 1965, although a special case was used in 1900 by Gordan, Hilbert's "rival" in the classical theory of invariants) has reinvigorated algebraic geometry in our times by providing tools to easily and naturally compute with ideals.
phoenixthoth said:
Is it differential Galois theory in which they prove certain elementary functions don't have elementary antiderivatives?
"Differential Galois theory" is a somewhat ambiguous phrase, but, yes, the theory developed by mathematicians such as Liouville, Picard, Vessiot, and Lie, which shows that certain elementary indefinite integrals cannot be expressed in terms of elementary functions, was directly inspired by Galois theory. This theory concerns what we could call the Galois theory of a special kind of ordinary differential equation, a
Fuchsian equation, in which we exploit the algebraico-geometric structure of certain spaces of functions. Here, the
monodromy group plays a role analogous to the Galois group in Galois theory; in fact, as mathwonk mentioned, sometimes one can use one to compute the other.
Liouville, BTW, was the mathematician who was probably the first person other than Galois himself to understand what Galois had done, and fortunately he took the time to explain it to others, by writing the first algebra textbook containing an exposition of Galois theory! This book had a life-altering impact on both Klein and Lie. I should add that Lie is responsible for (mostly rather different) development involving differential equations which was also directly inspired by "the spirit of Galois theory". As for Fuchs, his name is absent from the above list because according to legend, Poincare was once so annoyed by a mathematician by this name who claimed credit for something HP had done, that he named something completely different after Fuchs, just to embarrass poor Fuchs, who was obliged to spend the rest of his life disavowing any knowledge of "Fuchsian equations"!
Even more important was the effect of Galois theory of Klein and Lie, who worked together very closely c. 1870, soon after Galois theory had been rediscovered and explained to other mathematicians by Liouville. Indeed, modern scholars feel that the Erlangen program should perhaps be viewed as a collaboration between Klein and Lie. While Klein devoted much of the rest of his career to working out examples of the correspondence between geometries and "invariants" of group actions, Lie was particularly inspired by the idea of solving (systems of) ordinary and partial differential equations by finding and exploiting their "internal symmetries". He spent the rest of his career working out this great idea, with some necessary detours to develop what we know call the theory of Lie groups and Lie algebras, which was required background material which didn't exist, so Lie had to invent it. I hardly need explain the important of Lie theory to modern mathematics and physics, but I will say a bit more about Lie's theory of symmetry of differential equations.
This theory falls rather neatly into two parts. In the case of systems of ODEs, Lie showed how we can find internal symmetries and exploit them to reduce the order of the ODEs, say from second order to first order (classically, this process is called "finding a
first integral"). In the case of systems of PDEs, Lie showed how we can find internal symmetries (remarkably, while the original system can be highly nonlinear, Lie's equations determining the Lie algebra of symmetries is linear) and exploit them to reduce the number of variables, perhaps even turning a system of nonlinear PDEs into a system of ODEs, which can then be attacked using the first part of his theory.
Lie's theory of symmetries of ODEs underlies most of what symbolic computation packages like Maple do when they solve differential equations symbolically. As Lie showed, most techniques of solving ordinary differential equations which students learn even today in "cookbook courses" are in fact examples of exploiting symmetries of ODEs. And Lie's theory of symmetries of PDEs provides essentially the only known general approach to nonlinear PDEs. It has proven particularly useful in the theory of solitons. I might mention that it has a close connection with Noether's theorem relating symmetries of PDEs to conserved quantities, which allows us, given say a nonlinear "wave equation", to write down notions of "energy", "momentum" and perhaps additional quantities, which will be "conserved" in the sense that the energy of solution, evaluated at some time, remains valid for all later times.
All of these things flowed from just two papers by Galois (or three, depending upon how you count), both shorter and even more cryptic than most arXiv eprints. It's little wonder that the first people to read them, including the great mathematician Cauchy, were utterly baffled by what Galois was trying to say. Fortunately, Liouville was able to figure it out, once he had redisovered some of the ideas first hinted at by Galois in his 1831 paper.
Mathwonk already took a stab at another good question: "what kind of symmetry of the roots of a polynomial is captured by the Galois group?" He sketched very nicely the basic ideas of the
discriminant of a polynomial, i.e. the product of the pairwise differences of the roots
\Delta = \prod_{1 \leq i < j \leq n} \left( x_i - x_j \right)
This is a very important example of a
polynomial invariant of a finite matrix group. In embryonic form, we can already see here the essential ideas of the theory of (polynomial) invariants of
reductive groups.
Consider the symmetric group S_n. One can "linearize" its natural action on a set of n elements by representing each permutation as an n by n matrix whose rows (columns) each contain exactly one 1 with the other entries being 0. As was shown by Newton, the ring of polynomial invariants of this action, i.e. the ring of
symmetric polynomials, is generated by the
elementary symmetric polynomials:
f_1 = x_1 + x_2 + \dots x_n
f_2 = x_1 \, x_2 + x_1 \, x_3 + \dots x_{n-1} \, x_n
through
f_n = x_1 \, x_2 \dots x_n
More generally, the invariant ring for any subgroup of the symmetric group (has a direct decomposition as the sum of subspaces of homogeneous invariants of degree d, for each nonnegative integer d. The
Hilbert series counting the dimension of these vector subspaces of the ring of polynomial invariants of a matrix group can be computed by a handy formula due to Molien, which involves simply averaging certain expressions over the permutation matrices in the group. (This is related to averaging sizes of fixsets--- if you know about the Cauchy-Burnside formula in elementary group theory, this should sound familiar.) For S_3 we obtain from
Molien's formula
\frac{1}{(1-z) \, (1-z^2) \, (1-z^3)} <br />
= 1 + z + 2 \, z^2 + 3 \, z^3 + \dots
and for A_3 we obtain
\frac{1+z^3}{(1-z) \, (1-z^2) \, (1-z^3)} <br />
= 1 + z + 2\, z^2 + 4 \, z^3 + \dots
So in the case of A_3, we have an "extra" homogeneous polynomial invariant in the vector basis for the subspace of homogeneous invariants of degree three. We can take this to be the discriminant
<br />
\Delta(x_1,x_2,x_3) <br />
= \left( x_1-x_2 \right) \, \left( x_2 - x_3 \right) \, \left( x_1 - x_3 \right)
Moreover, writing the invariant ring of S_3 as R and that of A_3 as S, we have
S = R \oplus \Delta \, R
which adumbrates the all-important
Cohen-Macaulay property.
Similarly, the other subgroups of the symmetric group have invariant rings for which in the Molien series, the numerator lists (additively) the number of
secondary invariants by degree, while the denominator lists (multiplicatively) the number of
primary invariants by degree. The primary invariants are the generators of R, namely the elementary symmetric polynomials. A fundamental fact proven by Hilbert is that the there exist no algebraic relations among the primary invariants (no polynomial function of them vanishes identically), but the secondary and primary invariants do admit such relations, which were called
syzygies by Sylvester. A fundamental task of
commutative algebra is to compute the syzygies (and the syzygies of the syzygies--- another of Hilbert's great theorems states that this process terminates after finitely many steps). Noether's
transfer operator is a R-module homomorphism from the R-module S to R, and in each dimension its kernel gives the polynomials which are invariant under the subgroup but not the full symmetric subgroup. The Cohen-Macaulay property is essentially the fact that S is a finitely generated free R-module.
With minor changes, these facts are true for other
reductive groups, including
reflection groups.
In the case of Galois groups, in the generic case the only invariants are the primaries. We don't know the roots but we do know that the values assumed by the primaries when we plug in the roots for (x_1, x_2, \dots x_n) are simply the coefficients of the polynomial, which are rational. Sometimes the Galois group is smaller, a proper subgroup of the symmetric group. That happens when the polynomial factors (in which case the Galois group doesn't act transitively). More interesting is the possibility that the Galois group is a transitive subgroup of the symmetric group, which happens if the polynomial is irreducible but has some "extra invariants" corresponding to "unexpected asymmetries" in (algebraic combinations of) the roots. In this case, the extra invariants give certain polynomial combinations of the (unknown) roots which happen to be rational.
This is particularly easy to understand in the case of the discriminant, where the syzygy is a quadratic expression in the discriminant (secondary invariant) and the primary invariants. To wit, the square of the discriminant is a certain polynomial function of the primary invariants (or if you prefer, of the coefficients of the polynomial we are investigating). More generally, one can write down syzygies for the invariant rings of the other transitive subgroups. Plugging in the value of the unknown roots, we obtain polynomial expressions (incorporating the coefficients of the polynomial) which may be reducible; if so, we have found an extra invariant. Combining such information about the ring of invariants, one can try to determine the Galois group directly. This is the essential idea behind Newton's method of computing Galois groups: find a collection of
witnesses, interrogate them, and figure out who must be reponsible for the pattern of evidence so obtained. (In the modern literature, this idea of Newton has developed into a circle of ideas using
Schubert polynomials, a generalization of
Schur polynomials, which give a basis of the vector subspace of homogeneous symmetric polynomials of degree d, as in the Hilbert decomposition of the invariant ring of the symmetric group.)
Sturmfels disavows Newton's approach in his book, without explanation, but I guess the reason may be that computing syzygies quickly becomes notoriously demanding on computational resources; other approaches to computing the Galois group are far more efficient. Still, Newton's way of understanding Galois groups is perhaps the most direct route to understanding, and it fits them into a rather grand picture.
Returning to the Kleinian theme, one can now attempt to interpret the algebraic relations implicit in our polynomial invariants in geometric terms. In his book, Sturmfels discusses this in some detail for complex projective spaces and for euclidean spaces, but there is much, much more to say.
