An Introduction to p-adic Numbers and p-adic Analysis by Andrew Baker

By Andrew Baker

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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.

The open disc centred at α of radius δ is D (α; δ) = {γ ∈ Qp : |γ − α|p < δ}. The closed disc centred at α of radius δ is D (α; δ) = {γ ∈ Qp : |γ − α|p δ}. Clearly D (α; δ) ⊆ D (α; δ). Such a notion is familiar in the real or complex numbers; however, here there is an odd twist. 2. Let β ∈ D (α; δ). Then D (β; δ) = D (α; δ) . Hence every element of D (α; δ) is a centre. Similarly, if β ′ ∈ D (α; δ), then D (β ′ ; δ) = D (α; δ). Proof. This is a consequence of the fact that the p-adic norm is non-Archimedean.

M. Robert, A course in p-adic analysis, Springer-Verlag, 2000. 53 Problems Problem Set 1 1-1. For each of the following values n = 19, 27, 60, in the ring Z/n find (i) all the zero divisors, (ii) all the units and their inverses. 1-2. Let f (X) = X 2 − 2 ∈ Z[X]. For each of the primes p = 2, 3, 7, determine whether or not there is a root of f (X): (i) mod p, (ii) mod p2 , mod p3 , (iii) (iv) mod p4 . Can you say anything more? 1-3. Solve the following system of simultaneous linear equations over Z/n for each of the values n = 2, 9, 10: 3x + 2y − 11z ≡ n 7x + ≡ 12 2z − 8y + 1 n z ≡ n 2 1-4.

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