Local Galois module structure in positive characteristic and continued fractions
Let \(L/K\) be a cyclic extension of degree \(p\) of local fields of characteristic \(p\) with Galois group \(G\). Let \(L=K(y)\) where \(y^p-y=\alpha\) is given in normal form and \(\alpha\in K\) has valuation \(-t\). Let \(A\) and \(B\) be the rings of integers of \(K\) and \(L\) respectively. The main goal of this paper is to study the structure of \(B\) as \(R\)-module, where \(R=\{ x\in K[G]\mid xB\subseteq B\}\) is the associated order of \(B\). Let \(d\) be the minimal number of generators of \(B\) as \(R\)-module and let \({\mathfrak p}^{\delta}\) be the discriminant of \(B\) over \(A\). In [J. Number Theory 102, No. 1, 118--124 (2003; Zbl 1035.11059)] \textit{A. Aiba} studied the case \(d=1\), that is, when \(B\) is \(R\)-free. Aiba essentially proved that \(d=1\) if and only if \(s=p-1\) where \(s\) is the remainder on dividing \(t\) by \(p\). In the paper under review the authors prove that if \(p\mid\delta\) then \(B\) is free of rank~1 as \(R\)-module and if \(p\nmid \delta\) then the \textit{embedding dimension} \(e= \text{edim} (R)=\dim _ k ({\mathfrak m}/{\mathfrak m}^ 2)= 2d+1\), where \({\mathfrak m}\) is the maximal ideal of \(R\) and \(k=A/{\mathfrak p}\) is the residue field. In particular, if \(p\mid\delta\) then \(d=1\) and \(e=2\). This case includes the unramified case and also the case that the residue field extension is inseparable. Furthermore, \(R= A[G]\) if and only if \(L/K\) is unramified. The proof is done considering \(R\) as a graded ring and \(B\) as a graded \(R\)-module and then giving an explicit description of the gradings on \(R\) and \(B\) in terms of the balanced sequence associated to the fraction \(s/p\). The combinatorial description of the sequence also provides a method to compute \(d\). In the last section, the authors show how \(d\) can be computed more efficiently in terms of the continued fraction expansion of \(-s/p\).
Galois module structure, integral representation, local fields, Artin-Schreier extensions
scientific article; zbMATH DE number 5135806