Bases of associated Galois modules in general wildly ramified extensions and in elementary abelian extensions of degree 𝑝2

Abstract

The paper provides a comprehensive investigation of associated Galois modules and orders for totally ramified extensions of complete discrete valuation fields. The authors focus on explicit computations and systematic construction of bases for these modules, with particular emphasis on elementary abelian extensions of degree 𝑝2. The study introduces and develops the theory of graded-independent sets and diagonal bases, which enable constructive description of the modules 𝛾𝑖 and related associated orders. The central achievement is Theorem 3.3.2, which provides an explicit computation of the modules 𝛾𝑖 for extensions with Galois group (Z/ 𝑝Z)2 and ramification jumps distinct modulo 𝑝2. The paper thoroughly examines properties of the introduced constructions, including their relationship with classical associated orders and the behaviour under tame lifts. The obtained results are generalized to the case of relative associated modules 𝛾0
𝑖 = 𝛾𝑖 ∩𝑘0[𝐺], where 𝑘0 ⊂ 𝑘. The paper extensively utilizes the isomorphism between 𝐾 ⊗𝑘𝐾 and 𝐾[𝐺] constructed by the first author, and presents a detailed analysis of filtrations on tensor squares and their connection to Galois module structure. Respectively, the text can
be interesting to specialists in algebraic number theory and arithmetic geometry.