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We begin with the geometric problem of constructions with ruler and compass. We then introduce the notions of fields, of field extensions, and of algebraic extensions. This will quickly give us the key to the impossibility of some classical problems. In Chapter 5 we will be able to see how Galois theory gives a definitive criterion allowing us to decide if a geometric construction is, or is not, feasible with ruler and compass.

In the first chapter, the emphasis was on given numbers, and we were led to look at the equations of which they are solutions. In this chapter, we switch roles and look at polynomial equations and their eventual roots. Generalizing the construction of the field of complex numbers from the real numbers, we show how to create roots of a polynomial which does not have enough of them in a given field.

In this chapter we establish Galois correspondence. Discovered in 1832, it describes all subextensions of the spliting extension of a (separable) polynomial in terms of a subgroup of the group of permutations of the roots of this polynomial.

This chapter explains essential notions in group theory that one uses in Galois theory. This chapter was really taught according to the needs of students and should be read likewise.

We see in this chapter how Galois theory can be used to get a satisfactory answer to the problem of constructions with ruler and compass. By analogous methods, we discuss the problem of solving polynomial equations using radicals and we show how Galois theory allows us to understand the explicit resolution of equations of degrees up to 4. Finally, we will study the behavior of the Galois group of an equation when we vary the coefficients.

In this final chapter, I want to explain how certain aspects of the theory of linear differential equations with, say, polynomial coefficients, can be viewed in an algebraic setting. There is in fact a full “Galois theory of differential equations” of which I try to convey some ideas. I conclude with a theorem due to Liouville, a particular case of which is the fact that the function ∫ exp(x) dx has no elementary algebraic expression.