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

By Andrew Baker

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Katok, p-adic analysis compared with real, American Mathematical Society (2007). N. Koblitz, p-adic numbers, p-adic analysis and zeta functions, second edition, Springer-Verlag (1984). S. Lang, Algebra, revised third edition, Springer-Verlag (2002). K. Mahler, Introduction to p-adic numbers and their functions, second edition, Cambridge University Press (1981). [8] A. 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.

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 . 16. If 0 ̸= α ∈ Cp , then |α|p = 1 , pt where t ∈ Q. Proof. We know this is true for α ∈ Qalg p . By results of Chapter 2, if α = lim(p) αn n→∞ with αn ∈ Qalg p , then for sufficiently large n, |α|p = |αn |p . Next we can reasonably ask whether an analogue of the Fundamental Theorem of Algebra holds in Cp .

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. Let γ ∈ D (α; δ); then |γ − β|p = |(γ − α) + (α − β)|p max{|γ − α|p , |α − β|p } < δ. Thus D (α; δ) ⊆ D (β; δ). Similarly we can show that D (β; δ) ⊆ D (α; δ) and therefore these two sets are equal. A similar argument deals with the case of closed discs. Let X ⊆ Qp (for example, X = Zp ).

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