Elimination of maximal jumps

I. B. Zhukov ; O. Yu. Ivanova

Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 176-183

Résumé

This article continues a series of papers devoted to explicit constructions of Galois extension of complete discrete valuation fields of characteristic $0$ with the residue field of prime characteristic $p$, see [5], [6], [7], [8], [4], [10] and a survey article [9]. It is proved that any $p$-extension of a complete discrete valuation field containing a primitive $p$th root of unity can be embedded into a tower of Artin-Schreier extensions; an estimate for the height of this tower is obtained. This result also shows that such an extension can be embedded into Inaba extension, i. e., an extension obtained by the construction from [2]; an estimate for the order of the corresponding matrix is also obtained. Next, it is proved that any Galois $p$-extension of such field can be decomposed into a tower of Galois extensions of degree $p$ such that several upper levels have the maximal ramification jump whereas the lower ones are Artin-Schreier extensions.

Détail

Keywords: discrete valuation field, ramification jump, Artin-Schreier equation.