On the invariant factors of Kummer orders in the rings of integers of \(\mathfrak p\)-adic number fields of degree \(p^ 2\)
type: Item
instance of: scholarly article
title: On the invariant factors of Kummer orders in the rings of integers of \(\mathfrak p\)-adic number fields of degree \(p^ 2\)
zbMATH Open document ID: 0886.11066
author: Yoshimasa Miyata
published in: Journal of Number Theory
publication date: 1997-11-18T00:00:00Z
review text: If \(K=k(\root p \of u)\) is a Kummer extension of local fields, and \(u\in{\mathcal O}_k\) is a unit, then the so-called Kummer order \(\tilde{\mathcal O}_K={\mathcal O}_k[\root p \of u]\) is in general properly contained in \({\mathcal O}_K\). The finite quotient group is well-understood however; it is a function of either the ramification number, or of the discriminant, or of the level of the unit \(u\) (knowledge of any one out of these three suffices to determine \({\mathcal O}_K\)). Matters become naturally more complex for Kummer extensions \(K/k\) of degree \(p^2\). The author defines \(D(K)\) to be the sequence of elementary divisors of the quotient \([{\mathcal O}_K:\tilde{\mathcal O}_K]\). He develops criteria for \(D(K)=D(F)\) to hold, where \(F\) is another extension of the same kind. We will not go into the details because numerous case distinctions are necessary. (Remark: The paper first discusses the degree \(p\) case. The reviewer feels that the result in this case is known, but the only written account seems to be in the PhD thesis of \textit{M. Daberkow}, [Tech. Univ. Berlin (1995; Zbl 0867.11088)], and the formulation there is slightly different.) In particular the author shows that \(D(K)=D(F)\) may hold and yet \({\mathcal O}_K\) and \({\mathcal O}_F\) are not isomorphic. This could not happen in the degree \(p\) case. This kind of example, as well as another example of ``strange'' behavior, are obtained by combining the criteria obtained in this paper with the classification results of \textit{G. Elder} [Ann. Inst. Fourier 45, 625-647 (1995; Zbl 0820.11070)].
Mathematics Subject Classification ID: 11S23
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 1088557
zbMATH Keywords: Kummer extension of local fields
zbMATH Keywords: Galois modules
zbMATH Keywords: ramification
zbMATH Keywords: Kummer order
reviewed by: Cornelius Greither
MaRDI profile type: Publication
full work available at URL: jnth.1997.2170
OpenAlex ID: W2071856362
cites work: Q5532660
cites work: Q3922829
cites work: Galois module structure of ideals in wildly ramified cyclic extensions of degree \(p^ 2\)
cites work: The module structure of Kummer extensions over Dedekind domains.
cites work: Q3490098
cites work: On the Galois module structure of ideals and rings of all integers of \({\mathfrak p}\)-adic number fields
cites work: Wildly Ramified Gamma Extensions
DOI: 10.1006/JNTH.1997.2170
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description: scientific article; zbMATH DE number 1088557