" is a very stimulating text, which ... will attract mathematicians working in group theory, number theory, algebraic geometry, and complex analysis. -Zentralblatt fur 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"

• 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 fur 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"

• 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