By Alain M. Robert

ISBN-10: 1441931503

ISBN-13: 9781441931504

Came upon on the flip of the twentieth 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 deals many gains hardly handled in introductory p-adic texts comparable to topological versions of p-adic areas inside of Euclidian area, a unique case of Hazewinkel’s sensible equation lemma, and a therapy of analytic parts.

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

**Example text**

Its kernel is defined by kerr={XEQp:(x)EZ}. But (x) E Z means x = [x] + (x) E Zp, so that kerr = Z. The image of r consists of the complex numbers of the form exp(2rrik/pm) = exp(2Jri/pm)k. Since exp(27ri / pm) is a root of unity of order pm, these roots of unity generate when m varies among natural integers - the subgroup µ px. In particular, we have XEp-kZp = pkXEZp=r(X)pk=1 r(x)Eµpk. "a5 Comment. It is possible to give the factorization of rational numbers into pintegral and p-fractional components independent of the construction of p-adic numbers.

PROOF. Multiplying if necessary a p-adic expansion by a power of p, we see that it is enough to consider the case v(x) > 0, namely x E Zp. If the sequence (ai) is eventually periodic, x is the sum of an integer and a linear combination (with integral coefficients) of series of the form i` pS+ir = pS L. i>o 1 1-pt E Q and hence is a rational number. Conversely, suppose that x = F xi p' = alb is the p-adic expansion of a rational number (as we mentioned, we can assume that x E Zp; hence the summation is made for i > 0).

Then Ixy - abl = I(a+h)(b+k)-abl = Iak+hh - hkl + *
*

### A Course in p-adic Analysis (Graduate Texts in Mathematics) by Alain M. Robert

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