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On the main invariant of an element over a local field. (English) Zbl 0894.11045
Let \((K,v)\) be a discrete complete valued field and let \(v\) denote its unique extension to a fixed algebraic closure \(\overline{K}\) of \(K\). For \(a\in \overline{K}\setminus K\), the authors have already defined an invariant given by \[ \delta(a)= \sup \{v(a-c)\mid c\in \overline{K},\;[K(c):K]< [K(a):K]\}. \] They have proved that \(\delta(a)\) satisfies a fundamental principle like Krasner’s lemma satisfied by \(\omega(a)= \max\{v(a-a')\mid a'\neq a\) runs over \(K\)-conjugates of \(a\}\) [cf. J. Number Theory 52, 98-118 (1995; Zbl 0838.11078)].
In this paper, to \(a\) in \(K^{\text{sep}}\) and \(\delta\) a real number, they have associated a subfield \(K(a,\delta)\) of \(K(a)\) which is defined to be the fixed field of the subgroup of \(\text{Gal} (\overline{K}/K)\) consisting of all those automorphisms \(\sigma\) for which \(v(a- \sigma a)>\delta\). Then \(K(a)= K(a, \delta(a))\), if and only if \(\delta(a)= \omega(a)\). The most crucial result is that \(K(a)/ K(a,\delta(a))\) is a wildly ramified extension. The authors also give a relation between \(\delta(a)\) and \(\Delta(a)= \min\{v(a'-a)\mid a'\neq a\) runs over \(K\)-conjugates of \(a\}\).

MSC:
11S15 Ramification and extension theory
12J10 Valued fields
11S20 Galois theory
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