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Extra info for An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes]

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In particular, Qp is a field. 31. Let R be field with norm N . Then R 26 ˆ not equal to {0}. Then N ˆ ({an }) ̸= 0. Put Proof. Let {an } be an element of R, ˆ ({an }) = lim N (an ) > 0. ), so for such an n we have an ̸= 0. So eventually an has an inverse in R. Now define the sequence (bn ) in R by bn = 1 if n M and bn = a−1 n if n > M . Thus this sequence is Cauchy and lim(N ) an bn n→∞ = 1, which implies that {an }{bn } = {1}. ˆ Thus {an } has inverse {bn } in R. 27 CHAPTER 3 Some elementary p-adic analysis In this chapter we will investigate elementary p-adic analysis, including concepts such as convergence of sequences and series, continuity and other topics familiar from elementary real analysis, but now in the context of the p-adic numbers Qp with the p-adic norm | |p .

The function ∥ ∥p : C(Zp ) −→ R+ is in fact a non-Archimedean norm on C(Zp ). 28. C(Zp ) is a ring with non-Archimedean seminorm ∥ ∥p . Moreover, C(Zp ) is complete with respect to this seminorm. 41 We do not give the proof, but leave at least the first part as an exercise for the reader. Now recall the notion of the Fourier expansion of a continuous function f : [a, b] −→ R; this is a convergent series of the form ) ∞ ( ∑ 2πx 2pix a0 + an cos + sin n n n=1 which converges uniformly to f (x). In p-adic analysis there is an analogous expansion of a continuous function using the binomial coefficient functions ( ) x x(x − 1) · · · (x − n + 1) Cn (x) = .

Prove that in Q3 , ∑ X pn pn ; ∑ nk X n with 0 ∞ ∑ 32n (−1)n n=1 42n n =2 ∞ ∑ 32n . X n ; ∑ Xn n . 3-10. For n 1, let X(X − 1) · · · (X − n + 1) n! and C0 (X) = 1; in particular, for a natural number x, ( ) x Cn (x) = . n Cn (X) = (a) Show that if x ∈ Z then Cn (x) ∈ Z. (b) Show that if x ∈ Zp then Cn (x) ∈ Zp . (c) If αn ∈ Qp , show that the series ∞ ∑ αn Cn (x), n=0 converges for all x ∈ Zp if and only if lim αn = 0. n→∞ ∑ n (d) For x ∈ Z, determine ∞ n=0 Cn (x)p . ∑ ∑ 3-11. (a) Let αn be a series in Qp .

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An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes] by Andrew Baker

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