On the invariant factors of Kummer orders in the rings of integers of \(\mathfrak p\)-adic number fields of degree \(p^ 2\)
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)].
scientific article; zbMATH DE number 1088557
scientific article