Preface

Summary

The goal of the book is to lead the reader to an understanding of recent results on the Inverse Galois Problem: The construction of Galois extensions of the rational field ℚ with certain prescribed Galois groups. Assuming only a knowledge of elementary algebra and complex analysis, we develop the necessary background from topology (Chapter 4: covering space theory), Riemann surface theory (Chapters 5 and 6), and number theory (Chapter 1: Hilbert's irreducibility theorem). Classical results like Riemann's existence theorem and Hilbert's irreducibility theorem are proved in full, and applied in our context. The idea of rigidity is the basic underlying principle for the described construction methods for Galois extensions of ℚ.

From the work of Galois it emerged that an algebraic equation f(x) = 0, say over the rationals, is solvable by radicals if and only if the associated Galois group G_f is a solvable group. As a consequence, the general equation of degree n ≥ 5 cannot be solved by radicals because the group S_n is not solvable.

This idea of encoding algebraic–arithmetic information in terms of group theory was the beginning of both Galois theory and group theory. Nowadays we learn basic Galois theory in every first-year algebra course. It has become one of the guiding principles of algebra. One aspect of the theory that remains unsatisfactory is the fact that it is very hard to compute the Galois group of a given polynomial.