# zbMATH — the first resource for mathematics

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
Full Text: