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Now f ′ (2) ≡ 4 · 8 ≡ 2 and we can take u = 3. Then x = 2 − 3f (2) = −43 ≡ 7 is a root of f (X) 5 5 25 modulo 25. Repeating this we obtain 7 − 3f (7) = 7 − 75 = −68 ≡ 57 125 which is a root of the polynomial modulo 125. We now proceed as before. This method always works and relies upon Hensel’s Lemma (see Chapter 1 and Problem Set 3). 17 (Hensel’s Lemma). Let f (X) ∈ Zp [X] be a polynomial and let α ∈ Zp be a p-adic number for which f ′ (α) |f (α)|p < 1, p = 1. Define a sequence in Qp by setting α0 = α and in general αn+1 = αn − (f ′ (α))−1 f (αn ).

Earlier we saw that |π|p = p−1/2 , hence π is ramified. In fact we have e(α) = 2. 3. Now we can consider Qalg together with the norm | |p in the light of Chapter 2. It is p reasonable to ask if every Cauchy sequence in Qalg p has a limit with respect to | |p . 15. There are Cauchy sequences in Qalg p with respect to | |p which do not have limits. Hence, Qalg p is not complete with respect to the norm | |p . For an example of such a Cauchy sequence, see [5]. We can form the completion of Qalg p and its associated norm which are denoted Cp = Qalg p | | , p | |p .

I particularly recommend his discussion of p-adic integration, Γ-function and ζ-function. The world of p-adic analysis is in many ways very similar to that of classical real analysis, but it is also startlingly different. I hope you have enjoyed this sampler. We will now move on to consider something more like the complex numbers in the p-adic context. 45 CHAPTER 5 p-adic algebraic number theory In this section we will discuss a complete normed field Cp which contains Qp as a subfield and has the property that every polynomial f (X) ∈ Cp [X] has a root in Cp ; furthermore the norm | |p restricts to the usual norm on Qp and is non-Archimedean.

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