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Inhaltsverzeichnis

Frontmatter

1. Historical Aspects of the Resolution of Algebraic Equations

Abstract
In this chapter, we briefly recall the many different aspects of the study of algebraic equations, and give a few of the main features of each aspect. One must always remember that notions and techniques which we take for granted often cost mathematicians of past centuries great efforts; to feel this, one must try to imagine oneself possessing only the knowledge and methods which they had at their disposal. The bibliography contains references to some very important ancient texts as well as some recent texts on the history of these subjects (see, in particular, the books by J.-R Tignol and H. Edwards and the articles by C. Houzel).
Jean-Pierre Escofier

2. History of the Resolution of Quadratic, Cubic, and Quartic Equations Before 1640

Abstract
In this chapter, we give only a brief sketch of the rich history of low-degree equations; in particular, we have omitted the Indian and Chinese contributions. Readers interested in the subject can find excellent sources in the bibliography (see, in particular, the books by Tignol, Van der Waerden, and Yushkevich).
Jean-Pierre Escofier

3. Symmetric Polynomials

Abstract
In this chapter, we first give the basics on symmetric polynomials, and then present the notions of resultant and discriminant.
Jean-Pierre Escofier

4. Field Extensions

Abstract
In this chapter, we come to the basic notions of Galois theory. Abel and Galois defined the elements of a generated extension, but they did not envision these elements as forming a set. The concept of a field (and the word) did not appear until the work of Dedekind between 1857 and 1871. The abstract definition of a field was given about 20 years later by Weber and Moore. One hundred years ago, the language of linear algebra did not exist and results were formulated very differently from the way they are today, as can be seen, for example, in Weber’s book, listed in the bibliography.
Jean-Pierre Escofier

5. Constructions with Straightedge and Compass

Abstract
For the ancient Greeks, a geometric construction was a construction done using only straightedge and compass (a “straightedge” is a ruler not marked with any measurements). In this chapter, we consider planar problems, in the sense of elementary geometry. The verb “construct” means construct with straightedge and compass, according to the procedures described more precisely below.
Jean-Pierre Escofier

6. K-Homomorphisms

Abstract
In this chapter and the coming ones, we continue to restrict our attention to the situation of fields that can be realized as subfields of the field of complex numbers ℂ. However, the definitions and results all generalize directly to arbitrary fields contained in an algebraically closed field C of characteristic 0 (for fields of characteristic p ≠ 0, see Chapters 14 and 15).
Jean-Pierre Escofier

7. Normal Extensions

Abstract
Let K be a subfield of ℂ, and let P be a polynomial of degree n in K[X]. Let x1,...,x n be the (not necessarily distinct) roots of P in ℂ. The field K[x1,...,x n ], which by §4.6.2 is an extension of K of finite degree, is called the splitting field of P over K.
Jean-Pierre Escofier

8. Galois Groups

Abstract
In this chapter, we reach the very heart of Galois theory. To every polynomial with coefficients in a field K, with splitting field N over K, we associate a group G called its Galois group. We show that the subgroups of G are in bijective correspondence with the intermediate extensions between N and K. This correspondence makes it possible to solve problems about polynomials and their splitting fields algebraically, by computing groups. Over the following chapters, we sketch out this dictionary between the properties of an equation and the algebraic properties of its associated group.
Jean-Pierre Escofier

9. Roots of Unity

Abstract
In this chapter, we give the first example of a family of extensions whose Galois group is actually computable: this is the family of extensions of a field by roots of unity. The earliest work on this subject is due to Vandermonde (1770). It was followed by work of Gauss, in particular his beautiful discovery, on March 30, 1796, at the age of 19, of the construction of the regular polygon with 17 sides with ruler and compass (see Exercise 9.7) and its consequences.
Jean-Pierre Escofier

10. Cyclic Extensions

Abstract
After having studied extensions by roots of unity in Chapter 9, we now proceed to study extensions by roots of arbitrary elements of the base field, and consider in particular when such extensions have cyclic Galois group.
Jean-Pierre Escofier

11. Solvable Groups

Abstract
This chapter and the next one are devoted to the problem of resolving algebraic equations by radicals. Given a polynomial with coefficients in a field K, together with its splitting field N over K, the solvability of the equation P(x) = 0 by radicals can be expressed in terms of the existence of a particular sequence of intermediate extensions between K and N (see Chapter 12). By the Galois correspondence, this translates to a property of Galois group Gal(N|K). In this chapter, we introduce the groups having this special property, called solvability.
Jean-Pierre Escofier

12. Solvability of Equations by Radicals

Abstract
Using the correspondence constructed in the preceding chapters, together with group-theoretic results, Galois obtained his famous criterion of solvability by radicals. “This material is so entirely new that new names and new characters are necessary to express it,” he wrote, adding later that the true value of his criterion is essentially theoretical, as it is often impossible to compute the Galois group of a given polynomial: “In a word, the computations are not practicable.” However, he adds, the applications generally lead to “equations all of whose properties are known beforehand,” so that the computations are possible, as in Chapters 9 and 10.
Jean-Pierre Escofier

13. The Life of Évariste Galois

Abstract
The life of Évariste Galois is the most famous, fascinating, and commented life of any mathematician. It has even become something of a myth, like the lives of the immortal poets Rimbaud, Byron, or Keats.
Jean-Pierre Escofier

14. Finite Fields

Abstract
In this chapter, we drop the assumption that the fields we consider are subfields of ℂ. We will make use of analogues of some of the definitions and results of previous chapters, which adapt to the case of finite fields; we do not always give the new proofs for these results. Note, however, that Theorem 14.1.3 proves the existence of an algebraic closure for each of the fields we will study; it plays a role analogous to that of ℂ in the previous chapters. Fields of characteristic 2 are a particularly exciting subject of current research.
Jean-Pierre Escofier

15. Separable Extensions

Abstract
In this chapter, we consider arbitrary commutative fields, i.e. finite and infinite fields of arbitrary characteristic.
Jean-Pierre Escofier

16. Recent Developments

Abstract
Is every finite group the Galois group of an extension of the field ℚ? The answer to this question is not yet completely known.
Jean-Pierre Escofier

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