By Alain M. Robert

Chanced on on the flip of the 20 th century, p-adic numbers are usually utilized by mathematicians and physicists. this article is a self-contained presentation of uncomplicated p-adic research with a spotlight on analytic issues. It bargains many good points hardly taken care of in introductory p-adic texts akin to topological versions of p-adic areas within Euclidian house, a different case of Hazewinkel’s sensible equation lemma, and a therapy of analytic components.

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**Sample text**

Starting with the subring Z(p) C Q, we see that Zp appears also as a projective limit limZ(p)/p"Z(p) and hence as a completion of this ring Z(p). Comment. The presentation of the ring Zp of p-adic integers as a projective limit of the rings Z/p"Z shows that one can choose any system of representatives for Z mod pZ and write a corresponding expansion for any x E Zp in the form x = Y si pi with all digits Si E S. In particular, when the prime p is odd, it can also be useful to choose the symmetrical system of representatives S= 2 p1 2 ' 'C!

Consider now the cosets gH of H in G. Since translations are homeomorphisms of G, these cosets are open in G. Any union of such cosets is also open. But H is the complement of the union of all cosets gH ; H. Hence H is closed. 1. p-adic Numbers 20 Examples. The subgroups p"Zp (n > 0) are open and closed subgroups of the additive group Zp. The subgroups 1 + p"Zp (n > 1) are open and closed subgroups of the multiplicative group 1 + pZp. ,-T Let us recall that a subspace Y of a topological space X is called locally closed (in X) when each point y E Y has an open neighborhood V in X such that Y fl v is closed in V.

Y: KxK -> K, x i-> X-1 : K" -> K" are continuous. `^J ago 'v. fl via Unless explicitly stated otherwise, fields are supposed to be commutative. A topological field is a topological ring for which K" = K - (0} with the induced topology is a topological group. Equivalently, a topological field is a field K equipped with a topology such that T (x, y) H x - y is continuous on K x K, (x, y) H x/y is continuous on K" x K". Except for the appendix to Chapter II, we shall be interested only in valued fields: Pairs (K, I .