English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let p a prime number, Q_p the field of p-adic numbers, K a finite extensionof Q_p, \bar{K} an algebraic closure, and C_p the completion of Q_p, on whichthe valuation on Q_p extends. In his proof of the Ax-Sen-Tate theorem, Ax showsthat if x in C_p satisfies v(sx - x) > A for all s in the absolute Galois groupof K G, then there is a y in K such that v(x-y) >= A - C, with the constant C =p/(p-1)^2. Ax questions the optimality of this constant, which we study here.Introducing the extension of K by p^n-th roots of the uniformizer and relyingon Tate's and Colmez's works, we find the optimal constant and some moreinformation about elements in C_p satisfying v(sx - x) >= A for all s in G, wecompute the first cohomology group of G with coefficients in the ring ofintegers of \bar{K}.

Optimisation du th\'eor\`eme d'Ax-Sen-Tate et application \`a un calcul de cohomologie galoisienne p-adique