Based on a course given by the author, which focuses on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. This book is devoted to rationality and rigidity criteria and their application in realizing certain groups as Galois groups of regular extensions of Q(T).

Foreword, Notation, Introduction, 1 Examples in low degree, 2 Nilpotent and solvable groups as Galois groups over Q, 3 Hilbert’s irreducibility theorem, 4 Galois extensions of Q(T): first examples, 5 Galois extensions of Q(T) given by torsion on elliptic curves, 6 Galois extensions of C(T), 7 Rigidity and rationality on finite groups, 8 Construction of Galois extensions of Q(T) by the rigidity method, 9 The form Tr(x2) and its applications, 10 Appendix: the large sieve inequality, Bibliography

" is a very stimulating text, which . . . will attract mathematicians working in group theory, number theory, algebraic geometry, and complex analysis. —Zentralblatt für Mathematik This small book contains a nice introduction to some classical highlights and some recent work on the inverse Galois theory problem. The topics and main theorems are carefully chosen and composed in a masterly manner. —Mathematiacl Reviews -July 2007 ""Serre had the great good sense to have notes taken at his 1988 lectures at Harvard, creating a slim volume of great interest..."" -BOOK NEWS Inc., June 2008 J.-P. Serre, one of the greatest mathematicians in our time, provides here a unique introduction to both some classical milestones and some recent developments in the realm of inverse Galois theory. ... [This book] will maintain its unique, unparalleled role in the literature on inverse Galois theory for further generations. Now as before, J.-P. Serre's masterpiece of expository writing is an unvaluable source of inspiration and incitement likewise. -Werner Kleinert, Zentralblatt MATH, January 2007 ""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."" -MAA Reviews, September 2008"