On the Applications of Cyclotomic Fields in Introductory Number Theory

TL;DR

This paper explores how prime cyclotomic fields can be used to provide elegant proofs of number theory concepts, including a special case of Fermat's Last Theorem and modernized solutions to Pell's Equation.

Contribution

It introduces the concept of primary units in cyclotomic fields and demonstrates their role in proving key number theory results, offering new insights and methods.

Findings

Primary units are equivalent to real units in cyclotomic fields.

A proof of a special case of Fermat's Last Theorem is provided.

Dirichlet's solution to Pell's Equation is modernized.

Abstract

In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion of primary units in a cyclotomic field, demonstrate their equivalence to real units in this case, and show how this leads to a proof of a special case of Fermat's Last Theorem. We finally modernize Dirichlet's solution to Pell's Equation.