Witt Vector

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High Quality Content by WIKIPEDIA articles! Any p-adic integer can be written as a power series a0 + a1p1 + a2p² + ... where the a's are usually taken from the set {0, 1, 2, ..., p 1}. This set of representatives is rather artificial, and Teichmüller suggested the more canonical set consisting of 0 together with the p 1st roots of 1: in other words, the p roots of xp x = 0. These Teichmüller representatives can be identified with the elements of the finite field Fp of order p (by taking residues mod p), so this identifies the set of p-adic numbers with infinite sequences of elements of Fp. We now have the following problem: given two infinite sequences of elements of Fp, identified with p-adic numbers using Teichmüller's representatives, describe their sum and product as p-adic numbers explicitly. This problem was solved by Witt using Witt vectors.