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

By Alain M. Robert

ISBN-10: 1441931503

ISBN-13: 9781441931504

Stumbled on on the flip of the 20 th century, p-adic numbers are often utilized by mathematicians and physicists. this article is a self-contained presentation of uncomplicated p-adic research with a spotlight on analytic themes. It deals many beneficial properties infrequently taken care of in introductory p-adic texts resembling topological versions of p-adic areas inside of Euclidian house, a unique case of Hazewinkel’s practical equation lemma, and a remedy of analytic parts.

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A Course in p-adic Analysis (Graduate Texts in Mathematics) - download pdf or read online

Found on the flip of the 20 th 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 themes. It deals many gains infrequently handled in introductory p-adic texts similar to topological types of p-adic areas within Euclidian area, a distinct case of Hazewinkel’s useful equation lemma, and a remedy of analytic parts.

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

Example text

In fact, this inclusion is an equality. Indeed, if we take any b E H and assume (without loss of generality) b > 0, we can write b=ma+r (m EN, 0

Lyl, Ix + yl _< max(Ixl, IYI). In particular, we can define a metric on QP by d(x,y)=Ix-YI This distance satisfies d(x,y)>0ifx y and d(y, x) = d(x, y) coo as well as the triangle inequality in the strong ultrametric form d(x, y) < max(d(x, z), d(z, y)) <_ d(x, z) + d(z, y). coo This metric is invariant on the additive group d(x + z, y + z) = d(x, y) and also satisfies d(zx, zy) = Izi d(x, y) for all x, y, z E Q p. In particular, d(Px, PY) = d(x' Y) p From now on we shall always consider Q p as a metric field, endowed with this ultrametric distance.

With the p-adic metric the ring ZP is a topological ring. It is a compact, complete, metrizable space. PROOF. 1), it is enough to check the continuity of multiplication. Fix a and b in ZP and consider x = a + h, y = b + k in Z ,. Then Ixy - abl = I(a+h)(b+k)-abl = Iak+hh - hkl +

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


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