Topics in Galois Theory

Typically, after we prove the main theorems of Galois Theory in a course, we deduce the criterion for a polynomial to be solvable by radicals and then conclude by proving that the Galois group of the general polynomial of degree n over a function field on n variables is the symmetric group on n letters. One can think of this as the first instance of the inverse problem of Galois theory: Given a finite group G and a fixed field K, is there a Galois extension L of K such that its Galois group Gal(L/K) is isomorphic to the given group G?

The conjecture that the answer to this question is affirmative, attributed to Hilbert and Noether, remains open even in the original formulation when the base field is the rational numbers. For the case when G is the symmetric group on n letters, Hilbert proved in 1892 that if K is a number field, i.e., a finite extension of the rationals, there exists a Galois extension L of K such that its Galois group is isomorphic to the symmetric group. Hilbert's method is to use the general polynomial of degree n, whose coefficients are the symmetric functions s_i on the n variables x_i to solve the problem when the base field is the field of rational functions K(s₁, ..., s_n) and then show that the s_i can be specialized to elements of the base field such that the polynomial obtained with these substitutions has now coefficients in K, and the Galois group of its splitting field over K is isomorphic to the symmetric group.

Following these ideas Noether suggested a method to solve the inverse problem of Galois theory: By embedding the given group G in a permutation group, solve the problem for a purely transcendental extension of the given field K and then use Hilbert's irreducibility theorem to solve the problem over K. After Hilbert and Noether, the most important development was a theorem of Shafarevich that shows that all finite solvable groups are realizable as Galois groups of finite extensions of the rational numbers Q.

Serre's book focuses on this inverse problem of Galois theory, starting with some examples of groups of small order and then reviewing a theorem of Scholz and Reichardt on the realizability of p-groups, for p odd, as Galois groups over Q, or, equivalently, that every finite nilpotent group of odd order can be realized as a Galois group over Q. Since nilpotent groups are solvable, this theorem is a special case of Shafarevich's theorem.

Careful as Serre usually is, he remarks that in the original paper of Shafarevich there is a mistake regarding the prime 2, and in the notes of Shafarevich's Collected Papers (p. 752) there is a sketch of a method to correct it. In the second edition of Serre's book he quotes a complete solution to this problem in Chap. IX, Section 5 (p. 476-507) of the monograph Cohomology of Number Fields by Neukirch, Schmidt and Wingberg.

In Chapter 3 and 4, Hilbert's irreducibility theorem is proved and other methods from algebraic geometry are introduced, allowing Serre to realize Abelian, symmetric and alternating groups as Galois groups.

Galois extensions of Q(T) given by torsion points of elliptic curves and a theorem of Shih is the content of Chapter 5, and more algebraic geometry and topology are introduced in chapter 6 to treat Galois extensions of C(T) by viewing them as ramified coverings of the complex projective line and the corresponding Galois group as a fundamental group, and finally showing that the inverse Galois problem with base field C(T) has a solution in the affirmative for any finite group.

The remaining chapters treat the rationality and rigidity methods to construct Galois extensions of Q(T). Using these methods realizations of the symmetric, alternating, some projective special groups and some sporadic simple groups are obtained. The final chapters considers local criteria to formulate and prove some cases of the inverse problem of Galois theory, for example when the base field is a rational function field over the p-adic numbers, and using a result to Serre on the Hasse-Witt invariants of the quadratic form Tr(x²) to obtain the realization of central extensions of the alternating groups as Galois groups over certain fields.

Serre’s book helped to call the attention to a deep classical problem with connections to algebraic geometry, topology, algebra, and number theory. By carefully selecting examples, methods and topics, this book goes deeply into the problem. Despite its small size compares favorably with larger texts such as Inverse Galois Theory by G. Malle and B.H. Matzat (Springer 1999), Groups as Galois Groups by H. Volklein (Cambridge 1996) and Generic Polynomials: Constructive Aspects of the Inverse Galois Problem, Jensen, Ledet and Yui (Cambridge 2002).

For the second edition of the book some typographical errors have been corrected, e.g., the missing generator sigma in the first page of Chapter 1 of the first edition, the bibliography has been updated and some notes have been added regarding incomplete or non-published references at the time of publication of the first edition, e.g. the one already mentioned on Shafarevich’s theorem and one on p. 79 about the classification theorem of finite simple groups, specifically about “quasi-thin” groups, that was just published in 2004 (Aschbacher and Smith, The Classification of Quasi-Thin Groups, AMS).

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fzc@oso.izt.uam.mx