Three perspectives on p-adic numbers: analytic, algebraic, topological Nicolas Diaz-Wahl Abstract I outline the 3 main approaches to the construction of the p-adic numbers. The analytic approach constructs Qp by taking the completion of Q with respect to...
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Three perspectives on p-adic numbers: analytic, algebraic, topological Nicolas Diaz-Wahl Abstract I outline the 3 main approaches to the construction of the p-adic numbers. The analytic approach constructs Qp by taking the completion of Q with respect to the nonarchimedean absolute values, and classfies the nonarchimedean absolute values on Q as the p-adic ones. The algebraic approach defines completions with respect to topologies given by filtrations using inverse limits. Finally, Weil’s topological approach starts with a nonarchimedean local field in characteristic zero, and shows that such fields are precisely the p-adic fields. These three approaches give various insights into the p-adics. Local compactness is emphasized. 1 Introduction: Why p-adic numbers? I’ll start by motivating p-adic numbers. To do so, I’ll take a perspective that’s not any of those; instead think (algebro) geometrically. Take R = k[T1 , . . . , Tn ]. What does it mean to evaluate a polynomial? Take f (T ) ∈ R
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