Absolute Galois group

The absolute Galois group of a body is the Galois group , which belongs to the separable closure . It is unique except for isomorphism . In general, the expansion of the body is of infinite degree , which is why the main law of Galois theory is no longer applicable as such. The study of promises information about all finite Galois field extensions , in particular hints for solving the inverse problem of Galois theory . ${\displaystyle G_{K}}$ ${\displaystyle K}$ ${\displaystyle K_{sep}/K}$ ${\displaystyle K_{sep}/K}$ ${\displaystyle G_{K}}$ ${\displaystyle L/K}$

Examples

• For a perfect body , the separable closure is equal to the algebraic closure, so . ${\displaystyle K}$ ${\displaystyle K_{sep}={\overline {K}}}$
  • Because is , where the complex conjugation called. ${\displaystyle {\overline {\mathbb {R} }}=\mathbb {C} }$ ${\displaystyle G_{\mathbb {R} }=\{id,c\}}$ ${\displaystyle c}$
  • For no explicit characterization has been out of place. Statements It is hoped that from the set of Belyi , after the faithful on certain graphs , called dessins d'enfants , operates . The Absolute Galois group over the rational numbers is important in number theory and the subject of the Serre conjecture, which has since been proven . ${\displaystyle K=\mathbb {Q} }$ ${\displaystyle G_{K}}$ ${\displaystyle G_{K}}$
  • If the body is with elements, the following applies , with the projective limit of , the so-called test ring , being on the right-hand side . ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle q}$ ${\displaystyle G_{\mathbb {F} _{q}}=\lim _{\longleftarrow }\mathbb {Z} /n\mathbb {Z} =:{\hat {\mathbb {Z} }}}$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} ~(n\in \mathbb {N} )}$

literature

• Jürgen Neukirch: Algebraic number theory , Springer-Verlag.