# A Course in P-Adic Analysis

Language: English

Pages: 454

ISBN: 0387986693

Format: PDF / Kindle (mobi) / ePub

Kurt Hensel (1861-1941) discovered the p-adic numbers around the turn of the century. These exotic numbers (or so they appeared at first) are now well established in the mathematical world and used more and more by physicists as well. This book offers a self-contained presentation of basic p-adic analysis. The author is especially interested in the analytical topics in this field. Some of the features that are not treated in other introductory p-adic analysis texts are topological models of p-adic spaces inside Euclidean space, a construction of spherically complete fields, a p-adic mean value theorem and some consequences, a special case of Hazewinkel's functional equation lemma, a remainder formula for the Mahler expansion, and most importantly a treatment of analytic elements.

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the second one, *(H) = R/Z is the whole circle, Al in which case *,,(H) c R/p"Z must be a closed subgroup of finite index. Hence it must be open in this circle. By connectivity, *n(H) C R/pnZ. Since this must hold for all n > 1, we conclude that H = H = n f,.-, ((.fn H)) = Sp n>1 and H = SP in this case. (2) If *(H) = {0}, then H C >Ir-'(0) = Zp C Sp, and we have shown in (3.5) that the only possibilities are H = {0}, pkZp for some integer k > 0. DC, These possibilities occur in the list

tp((v;/vn)) = tp(w) E cp(Si). The remark made before proves that this scalar A satisfies Al I> 1, so that II(Ui)II. = IUnI = II < 1. This shows that v = g&,)) with 1: v E cp(B), where B = B<,(0, Qp). Consequently, B

center is closed and contains the rational field Q by assumption, hence is not finite. It is locally compact and not discrete by Corollary 2. 118 2. Finite Extensions of the Field of p-adic Numbers A.4. The Modulus is a Strict Homomorphism ;:' We claim that r' = m(K") is closed in R>o and m : K" IF is an open map. For r > 0, the compact set m(Br) is simply m(Br) = {0} U (I' fl [0, r]). In particular, if 0 < s < r < oc, r fl [s, r] is closed in R>o. Since the interiors of the intervals [e,

iterates that converges to 4. saw 6. Prove directly the following: If an -* 0 and bn -* 0 in an ultrametric field, then aibn-i -> 0 and cn Ea,, Ebn = Ecn. n>>0 n>0 n>0 [Hint. The assumption implies that the two sequences are bounded, say Jai 1 < C, Ibi I C for all i > 0, and for each given s > 0 there exists N = NE such that Jai l<_e, Ibil~~N). For i + j >_ 2N, we have I aibi 1 < cC, since one index at least is greater or equal to N.] 7. Show that two norms on a vector space define~~

Analytic Elements ..................... 354 Analyticity of Mahler Series ................... 354 *4.8 The Motzkin Theorem ...................... 357 Exercises for Chapter VI .......................... 359 7 Special Functions, Congruences 1. The Gamma Function F,, 1.1 Definition 366 ........ 366 ... ......................... 367 1.2 Basic Properties .......................... 368 Contents 1.3 The Gauss Multiplication Formula ............... 1.4 The Mahler Expansion ......... ...........