Ramified extensions of degree p and their Hopf–Galois module structure

Abstract

Cyclic, ramified extensions L/K of degree p of local fields with residue characteristic p are fairly well understood. Unless char(K) = 0 and L = K( p √ π K ) for some prime element π K ∈ K, they are defined by an Artin-Schreier equation. Additionally, through the work of Ferton, Aiba, de Smit and Thomas, and others, much is known about their Galois module structure of ideals, the structure of each ideal P n L as a module over its associated order A K[G] (n) = {x ∈ K[G] : xP n L ⊆ P n L } where G = Gal(L/K). This paper extends these results to separable, ramified extensions of degree p that are not Galois.

The ring of integers of Hopf-Galois degree p extensions of p-adic fields with dihedral normal closure

“…Remark 3.1. Following the terminology of [Eld18], in the case of a typical extension (a separable totally ramified degree p extension of local fields which is not generated by a p-th root of a uniformizer), this description of H is compatible with the one given in [Eld18, Theorem 3.1].…”

“…Recently, Del Corso, Ferri and Lombardo [CFL22] provided an alternative proof of the first three statements using the notion of minimal index of a Galois extension of p-adic fields. The third statement has also appeared as a particular case of [BCE18,Lemma 4.1] and [Eld18,Corollary 3.6] (in a slightly weaker form).…”

“…From [Ser79, Proposition 9] it is easily proved that this is always an integer number. Note that this number ℓ is different from the fractional ramification number denoted in [Eld18] in the same way; with our notation, that one is t 2 . As a result of the resemblance with the cyclic case, we will prove the following result, quite similar to Theorem 1.1.…”

“…We will be able to translate the arguments in [CFL22, Theorem 7.2] to prove Theorem 1.2 (2). On the other hand, our proof of Theorem 1.2 (3) will be a consequence of the work of Elder [Eld18] with typical degree p extensions, for which the criteria for the freeness depends on the construction of a scaffold. The last statement will be proved in Section 7 by characterizing when A θ is principal as A L/K -fractional ideal.…”

Hopf-Galois module structure of tamely ramified radical extensions of prime degree

“…However, the application of Hopf-Galois theory to extensions which are not Galois is particularly interesting, since it provides descriptions of rings of algebraic integers and/or fractional ideals in situations where classical techniques do not apply. For example, Hopf-Galois theory has recently been used by Koch [15] to study the structure of fractional ideals in a totally ramified purely inseparable extension of local fields of prime power degree, and by Elder [7] to address the same questions for a separable, but non-normal, ramified extension of local fields of prime degree. In this paper we study the Hopf-Galois module structure of the ring of algebraic integers in a tamely ramified non-normal radical extension of number fields of prime degree.…”