By Alain M. Robert

Found on the flip of the twentieth century, p-adic numbers are often utilized by mathematicians and physicists. this article is a self-contained presentation of simple p-adic research with a spotlight on analytic issues. It deals many gains not often handled in introductory p-adic texts corresponding to topological versions of p-adic areas within Euclidian area, a different case of Hazewinkel’s sensible equation lemma, and a remedy of analytic components.

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**Extra resources for A Course in p-adic Analysis (Graduate Texts in Mathematics)**

**Example text**

An interesting case is the following. Let G be a group and (Hn) a decreasing sequence of normal subgroups of G. We can then take Gn = G/Hn and (since 1. p-adic Numbers 32 cp : G/H the canonical projection homomorphism. The C projective limit of this sequence is a subgroup of the product G= lim G/H C O G/H E-together with the restrictions of projections ifr : G --* G/H,,. Since the system of quotient maps f, : G -p- G/H is always a compatible system, we get a factorizaG such that f = lJr o f. It is easy to determine the kernel of this tion f : G factorization f : f,,=nH,,.

5. Additive Structure of Qp and Zp Let us start with the sum formula Qp = Zp + Z[1/p] proved in the last section. Observe that this sum is not direct, since Zp n Z[1/p] = Z. fro oar The various embeddings that we have obtained are gathered in the following commutative diagrams giving the additive (resp. multiplicative) structure of Qp (resp. Qp ). Q Qp Z[1/p] Z[l/p] Zp QX QX P Zx (p) P z Z xP T N z P T N (1) (1) If we embed Z in the direct sum Zp ® Z[1/p] by means of m F-;, (m, -m) and call r the image, then the addition homomorphisms Z(p) ® Z[l/P] Z(p) + Z[l/P] = Q, Zp ® Z[l/p] - Zp + Z[l/p] = Qp have kernel IF and furnish isomorphisms (Z(P) (D Z[1/P]) / r = Q, (ZP ® Z[l/P]) / IF = Q.

Hence s = 1h, - hl E H and,-, -* 0. Since H is an additive subgroup, we must also have Z - 8 C H (for all n > 0), and the subgroup H is dense in R. 3. Topological Algebra 23 Corollary. (a) The only proper closed subgroups of R are the discrete subgroups aZ (a E R). C1+ (b) The only compact subgroup of R is the trivial subgroup (0}. (f' Using an isomorphism (of topological groups) between the additive real line and the positive multiplicative line, for example an exponential in base p R -+ R>0 t i-+ p`, (the inverse isomorphism is the logarithm to the base p) we deduce parallel results for the closed (resp.