From Polynomial Equations to Galois Groups

By Benjamin Schraen, Ecole Normale Supèrieure

In our modern world, our capacity to exchange large sets of highly secure data in a very short time has become essential. In this way, we have realized that some highly sophisticated methods in number theory are useful to produce some very secure and efficient cryptographic methods. However, number theory was studied for its own merit, far before the time of computers and a worldwide network. Algebraic numbers — or more comprehensively, solutions of polynomial equations — have an essential place in modern methods of cryptography. An essential tool for their study is the Galois group.

The Galois group was born from the answer to a question that has intrigued mathematicians for a long time: can we find a formula expressing the solutions of polynomial equations using only the coefficients of the equation and operations like addition, subtraction, multiplication, division and nth roots? Here is a more detailed description of the problem.

A mathematical equation is a relation between two quantities, one of which is often unknown. Solving the equation is to determine what values are allowed for this unknown quantity. In order to create such equations, we can use usual operations, like addition and multiplication. A polynomial equation is an equation in which only one unknown quantity x and its powers appear. For example, say we have 3x – 2 = 0, where the solution is the price of one item if the price for 3 is 2 euros; or say we have x² – 2 = 0, where the positive solution is the length of a square room from one corner to the opposite corner if one side is 2 square meters. Simple, isn't it? Not for long . . .

Mathematicians have searched for a simple and systematic formulation of the solutions of all polynomial equations for a long time. In the first example there is only one solution, which is x = 2/3. In the second example there are two solutions,  and . We are already forced to use a new operation, the square root. And in order to solve x³ – 5 = 0, we must again introduce cubic roots, and so on. We can ask if the use of all the nth roots is sufficient (with of course addition, multiplication, etc.) to solve all polynomial equations, knowing only the coefficients of the equation. For polynomial equations of degree 3 and 4, the answer is yes. But despite their work, mathematicians never found a general method to solve equations of degree 5, using only previous operations. And in fact, Evariste Galois, a French mathematician, proved in the 19th century that it is impossible to find a formulation involving only roots that could give solutions for polynomial equations of degree 5 or more. This result, already impressive, would not be so important if it also hadn't given us a very new and powerful tool that has now an important place in mathematics: the Galois group. A group is, not always but very often, a set of transformations of a space in which we can compose transformations. The set of plane transformations, which preserve distance, like rotations, is an example of this group; we call them isometries. In order to study the roots of a polynomial equation, Galois introduced the group consisting of permutations of all the solutions that preserve algebraic relations checked by these solutions. We call now this group the Galois group. Then Galois shed the light on the link between the algebraic properties of the solutions and the internal structure of the group, proving that in degrees 5 or more, the Galois groups can be much more complex than for small degrees. This is why it is impossible in general to describe the solutions using nth roots alone together with addition, subtraction, multiplication, division and coefficients of the equation. As we already suggested, the interest in this group ranges far from polynomial equations.

Number theory is a branch of mathematics that studies the comportment of integer solutions in certain problems. This kind of problem is often very difficult; Fermat's last theorem, solved in 1995, is a famous example. It states that for n greater than 3 there are no positive integers x, y and z such that xⁿ + yⁿ = zⁿ. However, it is often possible, by geometrical methods, to construct solutions to these problems . . . just not necessarily rational solutions. These solutions can be given by points, points whose coordinates are algebraic numbers (i.e., solutions of polynomial equations). Part of the difficulty is to determine which solutions are rational and which are integers in this bigger set. Sometimes a Galois group permutes this set of solutions so that it sufficiently perturbs these solutions, permitting us to recognize the ones that are given by rational numbers. They are the ones that are fixed by the Galois group! The Galois groups are surfacing in a large number of problems, ranging further and further away from simple polynomial equations; the knowledge of the structure of these Galois groups becomes a major issue for modern algebraic number theory.

In the last century deep links appeared between algebraic number theory, with its study of certain Galois groups, and some arithmetical objects coming from discrete structures in harmonic analysis. In fact, we do not have direct access to the absolute Galois group of the field of rational numbers. On the other hand, it seems far more accessible to study some finite dimensional spaces on which this group is acting as a transformation group; here we speak about Galois representations. A very efficient way to construct such representations comes from the theory of modular forms, which are functions invariant under some discrete groups of non-Euclidian isometries and which satisfy some particular differential equations. A question that highly interests mathematicians is this: Which Galois representations arise from the theory of modular forms? In the last ten years, conjugate works of many mathematicians led to an answer for two-dimensional representations. We can now understand how two-dimensional Galois representations and modular forms interact. But it is clear that to have a complete understanding of the Galois group of rational numbers, Galois representations of arbitrary dimension have to be investigated; the generalization of previous work will not be easy, because some new phenomena are appearing, but it will be well worth the effort.

 

Benjamin Schraen
Ecole Normale Supèrieure
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