On a theorem of Tignol for defectless extensions and its converse (Q2488310)

Let \(K\) be a field with a Henselian valuation \(v\), \(K ^ {\prime }\) a finite separable extension of \(K\), Tr\(_ {K'/K}\) the trace from \(K ^ {\prime }\) to \(K\), and \(v ^ {\prime }\) the unique valuation of \(K ^ {\prime }\) extending \(v\). We say that \(K ^ {\prime }\) is defectless over \(K\) (relative to \(v ^ {\prime }/v\)), if \([K ^ {\prime }\colon K] = [\widehat K ^ {\prime }\colon \widehat K]e(K ^ {\prime }/K)\), where \(\widehat K ^ {\prime }\) and \(\widehat K\) are the corresponding residue fields and \(e(K ^ {\prime }/K)\) is the ramification index of \(K ^ {\prime }/K\). It is easily seen that the set \(A _ {K'/K} = \{v(\text{ Tr}_{K'/K} (\alpha )) - v ^ {\prime } (\alpha )\colon \;\alpha \in K ^ {\prime \ast }\}\) consists of nonnegative elements of the value group \(v ^ {\prime } (K ^ {\prime })\). The paper under review proves that \(A _ {K'/K}\) has a minimum if and only if \(K ^ {\prime }/K\) is defectless. The validity of the right-to-left implication of this theorem in the special case where \([K ^ {\prime }\colon K]\) is a prime number has been known since 1990 [see \textit{J.-P. Tignol}, J. Reine Angew. Math. 404, 1--38 (1990; Zbl 0684.16011)]. For the proof of the main result, the authors show that if \(K ^ {\prime \prime }/K ^ {\prime }\) and \(K ^ {\prime }/K\) are defectless, then \(K ^ {\prime \prime }/K\) is defectless with \(A _ {K''/K} = A _ {K'/K} + A _ {K''/K'}\). They also give a new proof of the well-known result that finite extensions of formally \(p\)-adic fields (or more generally, of finitely ramified Henselian valued fields) are defectless.