Galois

Today is Évariste Galois’ 200th birthday.

The event will be celebrated with the publication of a new transcription and translation of Galois’ works (edited by Peter M. Neumann) by the European Mathematical Society. The announcement is here.

The life of Galois is such an incredible mixture of mathematics, romance, politics and human drama that, not surprisingly, he has been the subject of several biographies and even a film. Even now there are questions about his life and motivation to which generally agreed answers are not known. He was killed in a duel at the age of 20. (Was the duel the result of a love triangle, a frame-up by the Royalist police, or did Galois sacrifice himself for the republican movement in which he fervently believed?) His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost.

The occasion also gives me the opportunity to say a bit about Galois theory, and its place in mathematics.

Galois theory was, and still is, a theory of solving polynomial equations. The basic set-up is that f is a polynomial, with coefficients in a field K, which we wish to solve. The roots of f will generate a field L containing K. The degree of the extension L/K is the dimension of L regarded as a vector space over K. The larger the degree, the more work we have to do to find the roots.

Galois realised that solving an equation is essentially a process of breaking symmetry. If the polynomial is separable (a technical condition which holds in interesting cases, including those where K has characteristic zero or is a finite field), the group of symmetries of the roots of the equation (now called the Galois group of the equation) has order equal to the degree of L over K. The Galois group is a permutation group on the roots of the equation, and so has order at most n! for an equation of degree n.

So we can refine our earlier statement: it is not just the degree of the extension (the order of the Galois group) which measures how “hard” it is to solve the equation, but rather the structure of the group.

We are in the slightly unusual sitation where, the more symmetry the equation has, the more work we have to do to solve it. Indeed, a polynomial whose Galois group is the full symmetic group is the hardest to solve, while one whose Galois group is trivial is the easiest (its roots already lie in the field K.

An example is in order. Consider the polynomial f(x) = x²+1 over the real numbers R. This equation has two roots, commonly denoted by i and −i. But which is which? In other words, in the complex plane, which direction is up and which is down? There is no reason for preferring one root over the other; algebraically there is perfect symmetry between them. But we have to break the symmetry and make a choice in order to construct the complex number field C.

According to a theorem of Hilbert, the reverse of Leibniz’s dictum holds here: the worst of all possible worlds is also the most likely. That is, almost all polynomials of degree n (in a suitable sense) have Galois group S_n. A related open problem is whether this holds for particular classes of polynomials, for example:

Is it true that almost every irreducible polynomial (in a suitable sense) which occurs as a factor of the chromatic polynomial of a graph has symmetric Galois group?

Until the eighteenth century, the paradigm for solving polynomial equations was the formula for the solution of the quadratic: as well as the arithmetic operations of addition, subtraction, multiplication, and division, we are allowed to take square roots (or, in general, nth roots for arbitrary n). This last operation is necessary since the arithmetic operations don’t take us outside the field K.

By the end of the eighteenth century, equations of degree up to 4 had been solved in this way, i.e. “by radicals”, and it was suspected that equations of degree 5 cannot be so solved. This was proved by Abel in the early nineteenth century. But Galois did far more. His theory gives us a general criterion for deciding whether a particular polynomial can be solved by radicals:

Over a field of characteristic zero, a polynomial is solvable by radicals if and only if its Galois group is a soluble group (that is, has a series of normal subgroups with abelian quotients).

Indeed, this is why the term “soluble” (or, in American English, “solvable”) is used for groups with this property.

Over a field of prime characteristic p, we have to make a small variant of the process of solving by radicals: as well as the arithmetic operations, we are allowed to take nth roots if n is not divisible by p, and also to add solutions to equations of the form x^p = x+a for any element a of the field constructed so far. (Note that, if x is a solution of this equation, then the other solutions are x+1, x+2, …, x+p−1.)

The effect of Galois theory is much more far-reaching than this, and is felt in almost every part of mathematics that touches on modern number theory, and far beyond. Infinite Galois groups were one of the most important motivations for the flourishing theory of profinite groups; but the “absolute Galois group”, the group of symmetries of the algebraic numbers, is still one of the most mysterious objects in mathematics, being the subject of Grothendieck’s theory of dessins d’enfants, in which a childish scribble miraculously becomes a representation of the absolute Galois group.

One of the major unsolved problems of modern mathematics is the inverse Galois problem:

Is it true that, given any finite group G, there is a polynomial over the rational numbers whose Galois group is G?

At present, we have not much idea whether a breakthrough on this question will come from number-theoretic techniques or from our much improved understanding of finite groups following the classification of the finite simple groups.

The Galois group of the polynomial xⁿ−1 over the rational numbers is isomorphic to the multiplicative group of units in the integers modlo n; in particular, it is abelian. Conversely, Kronecker showed that an extension of Q with abelian Galois group is contained in a cyclotomic extension of Q. (This is not trivial. Say quickly: what is the smallest n for which the field generated by the roots of xⁿ−1 contains the square root of 2?)

The roots of unity which arise in this theorem are the values of the exponential function exp(2πiz) at rational values of z. Kronecker’s Jugendtraum (the “dream of his youth”) was that a similar relationship holds between abelian extensions of imaginary quadratic fields with values of elliptic functions. (Elliptic functions are so-called because they first arose in the calculation of the arc length of an ellipse. They are central in modern mathematics, arising in areas from Wiles’ proof of Fermat’s last theorem to the cryptography of chip and PIN payment systems.)

The resolution of the Jugendtraum in the twentieth century involved, and stimulated the development of, class field theory, by Tate and others. However, it seems that we do not know in detail what abelian extensions of real quadratic fields look like.

One piece of work which Galois did, which was published in his lifetime, is his construction of finite fields. The order of a finite field is necessarily a prime power; Galois showed that, for any given prime power, there is a field of that order, which is unique up to isomorphism. In his honour, we now call these fields Galois fields, and denote by GF(q) the field with q elements. Galois fields whose orders are large powers of 2 are important in the construction of error-correcting codes, shift registers, and other fundamentals of electronic engineering.