Calculation of norms of some secial elements of cyclotomic fields

TL;DR

This paper proves that a specific element is a unit in cyclotomic fields and establishes a connection between norms of certain elements and Lucas numbers for prime n, advancing understanding of algebraic integers in cyclotomic fields.

Contribution

It demonstrates that (1 - zeta + zeta^2) is a unit in the ring of integers of cyclotomic fields and links norms of quadratic polynomials in zeta to Lucas numbers for prime n.

Findings

(1 - zeta + zeta^2) is a unit in the ring of integers

Norm of (1 - zeta - zeta^2) equals the p-th Lucas number for prime n

Completes the study of norms of quadratic polynomials with coefficients ±1

Abstract

In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n, N_{Q(zeta)/Q}(1-zeta-zeta^2)=L(p) the p-th Lucas number thus completing the study of norms of quadratic polynomials in zeta that only have coefficients equal to 1 or -1 and both numbers appear.