By Andrew Baker

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13. The function ω : Zp −→ Qp is locally constant and satisﬁes the conditions ω(αβ) = ω(α)ω(β), |ω(α + β) − ω(α) − ω(β)|p < 1. Moreover, the image of this function consists of exactly p elements of Zp , namely the p distinct roots of the polynomial X p − X. Proof. The multiplicative part follows from the deﬁnition, while the additive result is an easy exercise with the ultrametric inequality. For the image of ω, we remark that the distinct numbers in the list 0, 1, 2, . . , p − 1 satisfy |r − s|p = 1.

2)n Cn (x). n=0 Of course, this is just the binomial series for (1 − 2)x in Q2 . The exponential and logarithmic series. In real and complex analysis the exponential and logarithmic power series exp(X) = log(X) = ∞ ∑ Xn n! n=0 ∞ ∑ , (−1)n−1 n=1 Xn n are of great importance. We can view each of these as having coeﬃcients in Qp for any prime p. The ﬁrst issue is to determine the p-adic radius of convergence of each of these series. Further details on this material can be found in [5]. 11, the p-adic radii of convergence of the p-adic power series expp (X) = ∞ ∑ 1 n X , n!

29, we know that for α ∈ Zp , there is a p-adic expansion α = α0 + α1 p + · · · + αn pn + · · · , where αn ∈ Z and 0 αn (p − 1). Consider the functions fn : Zp −→ Zp ; fn (α) = αn , which are deﬁned for all n 0. We claim these are locally constant. To see this, notice that fn is unchanged if we replace α by any β with |β − α|p < 1/pn ; hence fn is locally constant. We can extend this example to functions fn : Qp −→ Qp for n ∈ Z since for any α ∈ Qp we have an expansion α = α−r p−r + · · · + α0 + α1 p + · · · + αn pn + · · · and we can set fn (α) = αn in all cases; these are still locally constant functions on Qp .

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

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