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By Andrew Baker

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

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Then define N (f (X)) = e− ordX f (x) . 4 with ordX in place of ordp . Hence deduce that N satisfies the conditions required to be a non-Archimedean norm on R. 56 3-2. Which of the following are Cauchy sequences with respect to the p-adic norm | |p where p is a given prime? , (c) xn (this depends on x), (d) ap (this depends on a), (e) ns for s ∈ Z (this depends on s). In each case which is a Cauchy sequence find the limit if it is a rational number. 3-3. Let f (X) ∈ Z[X] and let p be a prime. , f (a0 )≡0).

First we define the Teichm¨ n Let α ∈ Zp ; then the sequence (αp ) is a sequence of p-adic integers and we claim it has a limit. To see this, we will show that it is Cauchy and use the fact that Qp is complete. 29, α has a unique p-adic expansion α = α0 + α1 p + α2 p2 + · · · with αk ∈ Z and 0 αk (p − 1). In particular, |α − α0 |p < 1. 26, in Z we have α0p ≡ α0 , p hence |α0p − α0 |p < 1. Making use of the fact that αk α0p−1−k p 1 together with the triangle inequality, we obtain |αp − α0p |p = (α − α0 )(αp−1 + αp−2 α0 + · · · + α0p−1 ) p |α − α0 |p < 1.

Let (αn ) be a sequence in Qp . 1. (αn ) is a Cauchy sequence in Qp if and only if (αn+1 − αn ) is a null sequence. Proof. See Problem set 3. Next we will now consider series in Qp . Suppose that (αn ) is a sequence in Qp . For each n ∑ we can consider the n-th partial sum of the series αn , s n = α1 + α2 + · · · + αn . 2. If the sequence (sn ) in Qp has a limit S = lim(p) sn we say that the series ∑ n→∞ αn converges to the limit S and write ∞ ∑ S is called the sum of the series ∑ αn = S. n=1 αn .

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

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