\contentsline {chapter}{Foreword}{ix}
\contentsline {chapter}{Notation}{xi}
\contentsline {chapter}{Introduction}{xiii}
\contentsline {chapter}{\numberline {1}Examples in low degree}{1}
\contentsline {section}{\numberline {1.1}The groups ${\bf Z}/2{\bf Z}$, ${\bf Z}/3{\bf Z}$, and $S_3$}{1}
\contentsline {section}{\numberline {1.2}The group $C_4$}{2}
\contentsline {section}{\numberline {1.3}Application of tori to abelian \hfil \penalty -\@M Galois groups of exponent $2,3,4,6$ }{6}
\contentsline {chapter}{\numberline {2}Nilpotent and solvable groups as Galois groups over ${\bf Q}$}{9}
\contentsline {section}{\numberline {2.1}A theorem of Scholz-Reichardt}{9}
\contentsline {section}{\numberline {2.2}The Frattini subgroup of a finite group}{16}
\contentsline {chapter}{\numberline {3}Hilbert's irreducibility theorem}{19}
\contentsline {section}{\numberline {3.1}The Hilbert property}{19}
\contentsline {section}{\numberline {3.2}Properties of thin sets}{21}
\contentsline {subsection}{\numberline {3.2.1}Extension of scalars}{21}
\contentsline {subsection}{\numberline {3.2.2}Intersections with linear subvarieties}{22}
\contentsline {section}{\numberline {3.3}Irreducibility theorem and thin sets}{23}
\contentsline {section}{\numberline {3.4}Hilbert's irreducibility theorem}{25}
\contentsline {section}{\numberline {3.5}Hilbert property and weak approximation}{27}
\contentsline {section}{\numberline {3.6}Proofs of prop.\ \ref {prop:proposition1} and \ref {prop:proposition2}}{30}
\contentsline {chapter}{\numberline {4}Galois extensions of Q(T): first examples }{35}
\contentsline {section}{\numberline {4.1}The property Gal$_T$}{35}
\contentsline {section}{\numberline {4.2}Abelian groups}{36}
\contentsline {section}{\numberline {4.3}Example: the quaternion group $Q_8$}{38}
\contentsline {section}{\numberline {4.4}Symmetric groups}{39}
\contentsline {section}{\numberline {4.5}The alternating group $A_n$}{43}
\contentsline {section}{\numberline {4.6}Finding good specializations of $T$}{44}
\contentsline {chapter}{\numberline {5}Galois extensions of Q(T) given by torsion on elliptic curves}{47}
\contentsline {section}{\numberline {5.1}Statement of Shih's theorem}{47}
\contentsline {section}{\numberline {5.2}An auxiliary construction}{48}
\contentsline {section}{\numberline {5.3}Proof of Shih's theorem}{49}
\contentsline {section}{\numberline {5.4}A complement}{52}
\contentsline {section}{\numberline {5.5}Further results on ${\rm {\bf PSL}}_2({\bf F}_q)$ and ${\rm {\bf SL}}_2({\bf F}_q)$ as Galois groups}{53}
\contentsline {chapter}{\numberline {6}Galois extensions of C(T)}{55}
\contentsline {section}{\numberline {6.1}The GAGA principle}{55}
\contentsline {section}{\numberline {6.2}Coverings of Riemann surfaces}{57}
\contentsline {section}{\numberline {6.3}From ${\bf C}$ to ${\mathaccent "7016\relax {\bf Q}}$}{57}
\contentsline {section}{\numberline {6.4}Appendix: universal ramified coverings of Riemann surfaces with signature}{60}
\contentsline {chapter}{\numberline {7}Rigidity and rationality on finite groups}{65}
\contentsline {section}{\numberline {7.1}Rationality}{65}
\contentsline {section}{\numberline {7.2}Counting solutions of equations in finite groups}{67}
\contentsline {section}{\numberline {7.3}Rigidity of a family of conjugacy classes}{70}
\contentsline {section}{\numberline {7.4}Examples of rigidity}{72}
\contentsline {subsection}{\numberline {7.4.1}The symmetric group $S_n$}{72}
\contentsline {subsection}{\numberline {7.4.2}The alternating group $A_5$}{73}
\contentsline {subsection}{\numberline {7.4.3}The groups ${\rm {\bf PSL}}_2({\bf F}_p)$}{74}
\contentsline {subsection}{\numberline {7.4.4}The group ${\rm {\bf SL}}_2({\bf F}_8)$}{75}
\contentsline {subsection}{\numberline {7.4.5}The Janko group $J_1$}{77}
\contentsline {subsection}{\numberline {7.4.6}The Hall-Janko group $J_2$ }{78}
\contentsline {subsection}{\numberline {7.4.7}The Fischer-Griess Monster $M$}{78}
\contentsline {chapter}{\numberline {8}Construction of Galois extensions of Q(T) \hfil \penalty -\@M by the rigidity method}{81}
\contentsline {section}{\numberline {8.1}The main theorem}{81}
\contentsline {section}{\numberline {8.2}Two variants}{83}
\contentsline {subsection}{\numberline {8.2.1}First variant}{83}
\contentsline {subsection}{\numberline {8.2.2}Second variant}{84}
\contentsline {section}{\numberline {8.3}Examples}{85}
\contentsline {subsection}{\numberline {8.3.1}The symmetric group $S_n$}{85}
\contentsline {subsection}{\numberline {8.3.2}The alternating group $A_5$}{86}
\contentsline {subsection}{\numberline {8.3.3}The group ${\rm {\bf PSL}}_2({\bf F}_p)$}{87}
\contentsline {subsection}{\numberline {8.3.4}The ${\rm Gal}_T$ property for the smallest simple groups}{88}
\contentsline {section}{\numberline {8.4}Local properties}{88}
\contentsline {subsection}{\numberline {8.4.1}Preliminaries}{88}
\contentsline {subsection}{\numberline {8.4.2}A problem on good reduction}{89}
\contentsline {subsection}{\numberline {8.4.3}The real case}{90}
\contentsline {subsection}{\numberline {8.4.4}The $p$-adic case: a theorem of Harbater}{93}
\contentsline {chapter}{\numberline {9}The form Tr(x$^2$) and its applications}{95}
\contentsline {section}{\numberline {9.1}Preliminaries}{95}
\contentsline {subsection}{\numberline {9.1.1}Galois cohomology (mod 2)}{95}
\contentsline {subsection}{\numberline {9.1.2}Quadratic forms}{95}
\contentsline {subsection}{\numberline {9.1.3}Cohomology of $S_n$}{97}
\contentsline {section}{\numberline {9.2}The quadratic form ${\rm Tr\,}(x^2)$}{98}
\contentsline {section}{\numberline {9.3}Application to extensions with Galois \hfil \penalty -\@M group ${\mathaccent "707E\relax A_n}$}{99}
\contentsline {chapter}{\numberline {10}Appendix: the large sieve inequality}{103}
\contentsline {section}{\numberline {10.1}Statement of the theorem}{103}
\contentsline {section}{\numberline {10.2}A lemma on finite groups}{105}
\contentsline {section}{\numberline {10.3}The Davenport-Halberstam theorem}{105}
\contentsline {section}{\numberline {10.4}Combining the information}{107}
\contentsline {chapter}{Bibliography}{109}
\contentsline {chapter}{Index}{117}