The formulas of coefficients of sum and product of p-adic integers with applications to Witt vectors

TL;DR

This paper derives explicit formulas for addition and multiplication of p-adic integers and applies these to compute Witt vector operations over finite fields, addressing coefficient transformation problems.

Contribution

It provides new explicit formulas for p-adic integer operations and extends these to Witt vectors over 4 and , solving coefficient transformation issues.

Findings

Explicit formulas for p-adic integer operations

Formulas for Witt vectors over 4 and 

Solution to Browkin's coefficient transformation problem

Abstract

The explicit formulas of operations, in particular addition and multiplication, of $p $-adic integers are presented. As applications of the results, at first the explicit formulas of operations of Witt vectors with coefficients in $\mathbb{F}_{2}$ are given; then, through solving a problem of Browkin about the transformation between the coefficients of a $p$-adic integer expressed in the ordinary least residue system and the numerically least residue system, similar formulas for Witt vectors with coefficients in $\mathbb{F}_{3}$ are obtained.