Tame Galois p -extensions of p -henselian fields

Model theory of finite-by-Presburger Abelian groups and finite extensions of $p$-adic fields

“…Since all a j are in particular in M K 2 , by Henselianity of K 2 and [5, Theorem 4.1.3, pp.88] we deduce that f has a root α in K 2 . Since by the first part, K 1 is relatively algebraically closed in K 2 , this α must lie in K 1 , and by another application of[5, Theorem 4.1.3, pp.88] we deduce that K 1 is Henselian. The proof of the Lemma is complete.…”

“…, where e is the ramification index and f the residue field dimension (see [5], [2]). Clearly it is a firstorder (but not yet visibly existential) property of O L (defined by ψ(x)) expressed in the language of rings that the residue field has p f elements.…”

“…Suppose K 1 is not relatively algebraically closed in K 2 , then K 1 (β) ⊂ K 2 , for some β which is algebraic over K 1 of degree m > 1. The valuation v of K 1 defined by ψ(x) has a unique extension w to K 1 (β) by Henselianity and [5,Theorem 4.4.1]. We have that m = e ′ f ′ , where e ′ is the ramification index and f ′ is the residue field dimension of K 1 (β) over K 1 with respect to w. (L satisfies all such equalities and so K 1 does too.…”

“…The valuation ring of the induced valuation on K 1 is K 1 ∩O K 2 , and its maximal ideal is M K 2 ∩ K 1 . By [5,Theorem 4.1.3,pp.88], Henselianity of a valued field is equivalent to the condition that any polynomial of the form…”

“…By Lemma 4, it is Henselian. Since any two Henselian valuation rings in K 1 are comparable, and K 1 has rank one value group (since its value group is a Z-group because it is elementarily equivalent to the value group of L), by [5,Theorem 4.4.1] the induced valuation on K 1 must agree with that given by O K 1 and 4.0.1 follows.…”