Fontaine and Wintenberger's theorem

The set of Fontaine and Wintenberger is a mathematical theorem in the field of algebra . He experienced an extensive generalization in Scholzes theory of perfectoid spaces .

Theorem : Let be a prime number, the field of the p-adic numbers and the field resulting from the adjunction of all iterated p-fold roots . Accordingly, let the body of the power series over the body with elements and the body resulting from the variable through the adjunction of all iterated p-fold roots . Then the absolute Galois groups of and are isomorphic to each other: ${\displaystyle p}$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle K=\mathbb {Q} _{p}(p^{\frac {1}{p^{\infty }}})}$ ${\displaystyle p}$ ${\displaystyle {\mathbb {F} }_{p}\left[[t\right]]}$ ${\displaystyle p}$ ${\displaystyle {\mathbb {F} }_{p}}$ ${\displaystyle K^{b}={\mathbb {F} }_{p}\left[[t\right]](t^{\frac {1}{p^{\infty }}})}$ ${\displaystyle t}$ ${\displaystyle K}$ ${\displaystyle K^{b}}$

${\displaystyle Gal({\overline {K}}/K)\simeq Gal({\overline {K^{b}}}/K^{b}).}$

literature

• J.-M. Fontaine , J.-P. Wintenberger : Extensions algébriques et corps des normes des extensions APF des corps locaux. CR Acad. Sci., Paris, Sér. A 288: 441-444 (1979).

Web links

• B. Bhatt: What is ... a perfectoid space? , Notice of the American Mathematical Society 10/2014, pp. 1082-1084.