On the Closed Subfields ofCp

Abstract

The so called ``p-adic analog of the field of complex numbers'' Cp (see [6, 7, 21]) seems to be a very interesting object to study, both from an algebraic and an analytic point of view. Particularly interesting are its closed subfields. A first account on them can be find in [7] (see also [14] and especially [21]). Here we try to give some new aspects and results on the elements of Cp which are transcendental over Qp . These elements will be called simply ``transcendental''. Our paper has six sections. In the first section we recall basic results, definitions and notations. In the second one we define ``distinguished sequences'' and prove that they permit to construct transcendental elements and also to associate to any transcendental element an infinite set of numerical invariants. Although the invariants of a given transcendental element do not define it uniquely, they tell much about it. In Section 3 we consider the conjugate class (or orbit) of a transcendental element. This orbit is compact and totally disconnected. In Theorem 3.7 we give an analytic criterion to determine the conjugates of a transcendental element. In Section 4 we deal with the so-called generic transcendental elements. Let K be a closed subfield of Cp , infinite over Qp . According to [13], there exists a generic transcendental element t of K, i.e. such that K is the topological closure of Qp(t). Starting with a distinguished sequence [:n]n which defines t we can describe the action of v on K, the residue field and value group of K and the algebraic closure of Qp in K (Theorems 4.1 and 4.2). In Section 5 we introduce a particular kind of generic transcendental elements. Although Theorem 5.4 gives a criterion for two such elements to Article No. NT972198