* * Member of the Roamnian Academy ON THE CONTINUITY OF THE TRACE

The notion of trace for an element in an algebraic extension of a field can be extended for (not necessary algebraic) elements of same extensions of a complete valued field (see [2]). However there are two different ex- tensions of such notion, namely, the trace as was defined in [7] and continuous trace defined in [2]. According to Example 4.3 in [2] there results that these two kinds of trace seem to be generally different. In this paper, we try to put in light same fundamental relations between there concepts of trace. 1. PRELIMINARIES 1. Let p be a prime number. As usual (see [A]) denote pQ the field of p-adic numbers and by ⋅ the p-adic module, normalized such that pp /1 =. Denote pQ a fixed algebraic closure of pQ and continue to denote by the same symbol ⋅ the unique extension of p-adic module to pQ. Furthermore, denote pC the completion of pQ with respect to module ⋅ and denote by the same symbol ⋅ the canonical extension of p-adic module to pC. 2. Denote ()pp QQGalG / = and consider on G the Krull topology. It is well know (see for example [APZ1]) that G is canonically isomorphic to ()ppcant QCGal / , the group of all continuous pQ authomorphism of pC. We shall assume ()ppcant QCGalG / =. One know that G, endowed with Krul