On saturated distinguished chains over a local field
Saturated distinguished chains were introduced by \textit{N. Popescu} and \textit{A. Zaharescu} [J. Number Theory 52, 98-118 (1995; Zbl 0838.11078)]. They used these chains to study the structure of irreducible polynomials in one variable over a local field \((K,v)\). From Krasner's lemma we have that for an algebraic separable element \(\alpha\), there exists an open neighborhood \(U\) of \(\overline K\), the algebraic closure of \(K\), such that \(K(\alpha)\subseteq K(\beta)\) for any \(\beta \in U\). This situation leads one to consider a sequence \((\alpha, \alpha _1,\ldots, \alpha _n)\) for \(\alpha\) such that \(\alpha _i\) is of minimal degree over \(K\) with \(\overline v (\alpha _{i-1}-\alpha _i)= \sup \{\overline v(\alpha _{i-1}-\beta)\mid \beta \in \overline K\) and \([K(\alpha _{i-1}): K]>[K(\beta):K]\}\) and \(\alpha _n\in K\). The chain \((\alpha, \alpha_1,\ldots, \alpha _n)\) is called a saturated distinguished chain (SDC). In the paper under review, the author considers SDC for an algebraic separable element over a local field. It is shown how those chains relate to the ramification for the field generated by these chains. After giving a review on SDCs, the case of tamely ramified extensions is treated. The elements appearing in an SDC for \(\alpha\) stay inside \(K(\alpha)\), and a concrete construction of an SCD is given for special cases. For wildly ramified extensions, the situation is different. In fact the next element \(\alpha_1\) to \(\alpha\) in a chain does not necessarily belong to \(K(\alpha)\). An explicit example is given such that \(\alpha _1 \not\in K(\alpha)\). In the last section, the author considers Galois wildly totally ramified extensions whose extension degrees are prime powers. Some results on the intermediate fields are obtained, but, in the general case, there are no concrete constructions of SDC, except for some cases.
scientific article; zbMATH DE number 1420252