On the p-th root of a p-adic number

TL;DR

This paper establishes a precise criterion for when a p-adic integer has a p-th root, demonstrates the falsity of Fermat's last theorem in p-adic contexts, and explores roots modulo powers of p.

Contribution

It provides a necessary and sufficient condition for p-th roots in p-adic integers and extends the discussion to residues modulo p^k, also showing Fermat's last theorem does not hold in these settings.

Findings

Characterization of p-th roots in p-adic integers

Fermat's last theorem is false for p-adic integers

Fermat's last theorem is false for residues mod p^k

Abstract

We give a sufficient and necessary condition for a p-adic integer to have p-th root in the ring of p-adic integers. The same condition holds clearly for residues modulo p^k. We give a proof that Fermat's last theorem is false for p-adic integers and for residues mod p^k.