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A trailer for p-adic analysis, second half: Mahler coefficients – Power Overwhelming

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In the previous post we defined -adic numbers. This post will state (mostly without proof) some more surprising results about continuous functions . Then we give the famous proof of the Skolem-Mahler-Lech theorem using -adic analysis. Before I go on, I want to mention that is not algebraically closed. So, we can take its algebraic closure — but this field is now no longer complete (in the topological sense). However, we can then take the completion of this space to obtain . In general, completing an algebraically closed field remains algebraically closed, and so there is a larger space which is algebraically closed and complete. This space is called the -adic complex numbers. We won’t need at all in what follows, so you can forget everything you just read. One of the big surprises of -adic analysis is that we can concretely describe all continuous functions . They are given by a basis of functions in the following way. Theorem 1 (Mahler; see Schikhof Theorem 51.1 and Exercise 51.B)

A trailer for p-adic analysis, second half: Mahler coefficients In the previous post we defined p p p -adic numbers. This post will state (mostly without proof) some more surprising results about continuous functions f ⁣ : Z p → Q p f \colon \mathbb Z_p \rightarrow \mathbb Q_p f : Z p ​ → Q p ​ . Then we give the famous proof of the Skolem-Mahler-Lech theorem using p p p -adic analysis. 1. Digression on C p \mathbb C_p C p ​ Before I go on, I want to mention that Q p \mathbb Q_p Q p ​ is not algebraically closed. So, we can take its algebraic closure Q p ‾ \overline{\mathbb Q_p} Q p ​ ​ — but

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