Witt Vector - Motivation

Motivation

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 not the only possible choice, and Teichmüller suggested an alternative 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 F_p of order p (by taking residues mod p), so this identifies the set of p-adic integers with infinite sequences of elements of F_p.

We now have the following problem: given two infinite sequences of elements of F_p, identified with p-adic integers using Teichmüller's representatives, describe their sum and product as p-adic integers explicitly. This problem was solved by Witt using Witt vectors.