On the Galois Module Structure of Ideals and Rings of All Integers of p-Adic Number Fields

Miyata, Yoshimasa

doi:10.1006/jabr.1995.1320

Cited by 7 publications

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“…Now we state results obtained in this paper. In Section 1, we recall a theorem from [8] giving an integral basis of O (Theorem 1) and prove that a quotient module MO/Õ is isomorphic to a direct sum of n copies of a quotient module o/oπ d , i.e. a homogeneous o-torsion module (Theorem 2).…”

Section: Introductionmentioning

confidence: 99%

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On the module structure of rings of integers in p-adic number fields over associated orders

Miyata

1998

Math. Proc. Camb. Phil. Soc.

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Let p be an odd prime number. Let k be a [pfr ]-number field and [ofr ] the ring of all integers of k with a prime element π. Let K/k be a cyclic extension with Galois group G, and [Afr ] the associated order of the ring [Ofr ] of all integers in K:[Afr ]={f∈kG[mid ]f[Ofr ]⊆[Ofr ]}.F. Bertrandias and M-J. Ferton [1] obtained necessary and sufficient conditions that [Ofr ] is [Afr ]-free in the case K/k is of degree p. The purpose of this paper is to study such conditions in case K/k is a cyclic totally ramified Kummer extension of degree n=pm.

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Section: Introductionmentioning

confidence: 99%

“…Let p = 3, m = 2 and k = Q 3 (ζ) with π = 1 − ζ. Let α = 3 √ π 9 √ 1 + π 8 and K = k(α). By [7, (3•9)], the second ramification number c 2 (i.e.…”

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confidence: 99%

On the module structure of rings of integers in p-adic number fields over associated orders

Miyata

1998

Math. Proc. Camb. Phil. Soc.

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“…Now we follow the way stated in [6,Section 5] and define an upper triangular matrix A(K)=(a h, l ) 0 h, l<p 2 as in the following. Let h= pj+i and l= pn+m with 0 i, j, m, n<p. As [:…”

Section: Isomorphism Classesmentioning

confidence: 99%

On the Invariant Factors of Kummer Orders in the Rings of Integers of p-adic Number Fields of Degreep2

Miyata

1997

Journal of Number Theory

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“…These extensions have been studied in a series of papers by Miyata[Miy95,Miy98,Miy04]. In particular, Miyata gave a necessary and sufficient condition in terms of b ≡ −t (mod q) for O E to be free over A E/F .…”

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Integral Galois module structure for elementary abelian extensions with a Galois scaffold

Byott

Elder

2014

Proc. Amer. Math. Soc.

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Abstract. This paper justifies an assertion in [Eld09] that Galois scaffolds make the questions of Galois module structure tractable. Let k be a perfect field of characteristic p and let K = k((T )). For the class of characteristic p elementary abelian p-extensions L/K with Galois scaffolds described in [Eld09], we give a necessary and sufficient condition for the valuation ring O L to be free over its associated orderInterestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.

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On the Galois Module Structure of Ideals and Rings of All Integers of p-Adic Number Fields

Cited by 7 publications

References 0 publications

On the module structure of rings of integers in p-adic number fields over associated orders

On the module structure of rings of integers in p-adic number fields over associated orders

On the Invariant Factors of Kummer Orders in the Rings of Integers of p-adic Number Fields of Degreep2

Integral Galois module structure for elementary abelian extensions with a Galois scaffold

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