Let \(K\) be a local field of characteristic \(p > 0\) with a perfect residue field \(k\) and let \(K _ s\) be a separable closure of \(K\). It is well known that \(k\) embeds in \(K\) as a subfield and \(K\) is \(k\)-isomorphic to the Laurent formal power series field \(k((\pi _ K))\) whenever \(\pi _ K\) is a uniform element of \(K\). We say that finite extensions \(L _ 1\) and \(L _ 2\) of \(K\) are \(k\)-isomorphic, if there exists a field isomorphism \(\sigma : L _ 1 \to L _ 2\), such that \(\sigma (K) = K\) and \(\sigma \) induces the identity map on \(k\).
The paper under review considers the problem of classifying the \(k\)-isomorphism classes of finite totally ramified extensions of \(K\). The tamely ramified case offers no difficulty – it is well known that tamely totally ramified extensions of \(K\) are simple radical, so it is easy to see that, for each \(n \in \mathbb N\), there exists a unique \(k\)-isomorphism class of degree \(n\) totally ramified extensions of \(K\). This paper turns to ramified extensions of \(K\) of degree \(p\), the simplest non-tame extensions.
The authors take as a starting point the observation that \(k\)-isomorphic extensions of \(K\) have the same ramification data. This allows them to consider the classification problem for totally ramified degree \(p\) extensions of \(K\) in \(K _ s\) of fixed ramification break \(b > 0\). Further, it becomes clear that the rational number \(b\) must satisfy the condition \((p - 1)b \in \mathbb N \setminus p\mathbb N\). When this condition holds, the authors fix a system \(R\) of coset representatives for the quotient group \(k ^ {\ast }/(k ^ {\ast }) ^ {(p-1)b}\), then they attach to each \(\omega \in R\) an Eisensteinian trinomial \(f _ {\omega }\) of degree \(p\), and fix a root \(\theta _ {\omega }\) of \(f _ {\omega }\) in \(K _ s\).
The main result of the paper states that the fields \(K(\theta _ {\omega })\), \(\omega \in R\), form a system of representatives of the \(k\)-isomorphism classes of totally ramified extensions of \(K\) in \(K _ s\) of degree \(p\) and ramification break \(b\). Furthermore, it gives a necessary and sufficient condition for an extension \(K(\theta _ R)/K\) to be Galois, a result obtained previously by B. Klopsch [J. Algebra 223, No. 1, 37–56 (2000; Zbl 0965.20021)]. The construction of \(f _ {\omega }\), \(\omega \in R\), is obtained by adapting similar results of S. Amano concerning totally ramified extensions \(E/F\) of degree \(p\), where \(F\) is a finite extension of the field \(\mathbb Q _ p\) of \(p\)-adic numbers see [J. Fac. Sci., Univ. Tokyo, Sect. I A 18, 1–21 (1971; Zbl 0231.12019)].