A p-ADIC EXAMPLE FOR THE CHARACTERIZATION OF THE CANONICAL p-HENSELIAN VALUATION

The authors have shown recently that the canonical p-henselian valuation is uniformly ∅-definable in the elementary class of fields which have characteristic p or contain a primitive pth root of unity ζp. In order to do this, we proved a classification of the canonical p-henselian valuation via case distinction. One of the cases discussed was previously not even known for algebraic extensions of the p-adic numbers. The aim of this note is to give a direct proof of this p-adic fact. In [JK14, Theorem 3.1], we give a uniform definition of the canonical p-henselian valuation. In this note, we show why one of the main ingredients involved ([JK14, Lemma 4.5]) holds for algebraic extensions of the p-adic numbers. The proof is completely independent of the one given in [JK14] and uses specific properties of the p-adics. However, it gave us reason to search for a proof of the more general Lemma. For an introduction to p-henselian valuations in general and the canonical p-henselian valuation in particular, see [EP05, §4.2], [Koe95] and [JK14]. We use the following notation: For a valued field (F, v), we denote the valuation ring by Ov and the maximal ideal by mv. Furthermore, we write Fv for the residue field and vF for the value group. The canonical p-henselian valuation on F is denoted by v F . The fact which we aim to prove in the p-adic context is the following: Lemma (Lemma 4.5 in [JK14]). Let (F, v) be a p-henselian valued field with char(F) = 0, char(Fv) = p and ζp ∈ F. Assume further that Fv is perfect, vF has a non-trivial pdivisible convex subgroup and that Fv = Fv(p) holds. Then, we have v = v F ⇐⇒ ∀x ∈ mv \ {0} : 1 + x (ζp − 1)Ov * (F×)p. The aim of this note is to show explicitely that the lemma holds for any algebraic extension F of Qp. For simplicity, we assume p > 2. On any such F there is a unique non-trivial (p-)henselian valuation, namely the unique extension vF of the p-adic valuation vp on Qp. Note that FvF is an algebraic extension of Fp and is thus perfect. Furthermore, the value group vF F has rank 1, i.e. it contains no non-trivial proper convex subgroups. Thus, in case the value group vF F contains a non-trivial p-divisible subgroup, it is already p-divisible. Hence, in the p-adic context, the lemma reads as follows: Lemma (p-adic version). Let p > 2 and (F, vF) be an algebraic extension of (Qp, vp) with ζp ∈ F, such that vF F is p-divisible and FvF = FvF(p) holds. Then, we have ∀x ∈ mvF \ {0} : 1 + x(ζp − 1)OvF * (F×)p. The rest of this note is a proof of the above Lemma. Take (F, vF) ⊇ (Qp, vp) an algebraic extension satisfying the conditions of the Lemma. In particular, F is an infinite algebraic extension of Qp. Assume for the sake of a contradiction that the condition in the Lemma does not hold, that is we have ∃x ∈ mvF \ {0} : 1 + x(ζp − 1)OvF ⊆ (F×)p. (1)