# Two classes of number fields with a non-principal Euclidean ideal

## TL;DR

This paper identifies two classes of totally real quartic number fields, including biquadratic and cyclic extensions, that possess non-principal Euclidean ideals, expanding understanding of Euclidean ideal classes in number fields.

## Contribution

It introduces new classes of totally real quartic number fields with non-principal Euclidean ideals, generalizing previous techniques used for specific cases.

## Key findings

- Identifies biquadratic and cyclic extensions with non-principal Euclidean ideals
- Generalizes Graves' techniques to broader classes of number fields
- Expands the catalog of number fields with known Euclidean ideal classes

## Abstract

This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that the number field $\mathbb{Q}(\sqrt{2},\sqrt{35})$ has a non-principal Euclidean ideal.