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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 .

Suppose also that f ′ (a0 ) is not congruent to 0 mod p. Show that the sequence p (an ) defined by an+1 = an − uf (an ), where u ∈ Z satisfies uf ′ (a0 )≡1, is a Cauchy sequence with respect to | |p converging to root of p f in Qp . 3-4. Let p be a prime with p≡1. 4 (a) Let c ∈ Z be a primitive (p − 1)-st root of 1 modulo p. By considering powers of c, show that there is a root of X 2 + 1 modulo p. (b) Use Question 3-3 to construct a Cauchy sequence (an ) in Q with respect to | |p such that a2n + 1 1 .

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 .

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


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