Euclidean algorithm in Galois Quartic Fields

TL;DR

This paper proves that all imaginary biquadratic and cyclic quartic fields with class number one are Euclidean, expanding understanding of Euclidean domains in algebraic number theory.

Contribution

It establishes that all such fields with class number one are Euclidean, a significant extension of known Euclidean properties in number fields.

Findings

All imaginary biquadratic fields are Euclidean.

All cyclic quartic fields with class number 1 are Euclidean.

Provides new classifications of Euclidean fields in algebraic number theory.

Abstract

We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.