Download e-book for kindle: A Course in p-adic Analysis (Graduate Texts in Mathematics) by Alain M. Robert

By Alain M. Robert

ISBN-10: 1441931503

ISBN-13: 9781441931504

Chanced on 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 simple p-adic research with a spotlight on analytic themes. It deals many positive aspects infrequently taken care of in introductory p-adic texts comparable to topological types of p-adic areas within Euclidian house, a different case of Hazewinkel’s sensible equation lemma, and a remedy of analytic parts.

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Let x = >J ai pi E Q, (i > v(x), 0 < ai < p - 1). , x E Q precisely when the sequence (ai) of digits of x is eventually periodic. 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).

Together with the theorem of the preceding section, this proposition establishes the following diagram of logical equivalences and implications for a topological group G and a subgroup H. 4. e==> H open 4 4 G/H Hausdorff H closed Closed Subgroups of the Additive Real Line Let us review a few well-known results concerning the classical real line, viewed as an additive topological group. At first sight, the differences with Zp are striking, but a closer look will reveal formal similarities, for example when compact and discrete are interchanged.

1). Theorem. The map E aiXe H >aipi : Z[[X]] ZP is a ring homomorphism. It defines a canonical isomorphism Z[[X ]]/(X - p) ZP, where (X - p) denotes the principal ideal generated by the polynomial X - p in the formal power series ring. PROOF Let us consider the sequence of homomorphisms fn : Z[[X]] -* Z/pnZ, >2aiX` - >aip` mod p". * limZ/p"Z = ZP compatible with the fn. If x = Y_ ai pi is any p-adic integer, then x = f (Y- ai X'), and this shows that f is surjective. We have to show that the kernel of f is the principal ideal generated by the polynomial X - p.

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


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