Continuous Functions on Zp
The goal of this chapter is the study of continuous functions on subsets of the p-adic field Q p with values in an extension of Q p. Since Q p admits a partition into clopen balls x + Z p (x ∈ Q p /Z p = Z[1/p]/Z), it is enough to study continuous functions on Z p. Thus, we shall typically study continuous functions Z p → C p . Since the natural numbers N form a dense subset of the ring Z p , we shall start by the study of functions on N or Z and with values in any abelian group.
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