Lifting polynomials over a local field (Q1885505)

In [J. Number Theory 52, No. 1, 98--118 (1995; Zbl 0838.11078)] \textit{N. Popescu} and the author studied the structure of irreducible polynomials in one variable over a local field \(K\). In that paper was defined the notion of satured distinguished chains of polynomials over \(K\). The knowledge of a satured distinguished chain of a given element \(\alpha \in \overline{K}\), where \(\overline{K}\) denotes a fixed algebraic closure of \(K\) is useful, among other things, for constructing integral basis of \(K(\alpha)\) over \(K\). The same work described a constructive way to produce all the irreducible polynomials in one variable over \(K\), via repeated operation of lifting. In the paper under review the author shows that the lifting process produces infinitely many lifting polynomials and studies the classification of these polynomials. In particular he investigates which lifting polynomials produce the same field extension of \(K\). This problem is considered when the lifting is done with respect to a general irreducible polynomial \(f\) over \(K\). Here \(K\) is considered of characteristic \(0\). After giving some general results and definitions, it is estimated the distance between the roots of a giving lifting polynomial. In the last part, the author considers two lifting polynomials obtained from the same initial data, and provides sufficient conditions under which these polynomials define the same field extension.