Euclidean algorithm in Galois quartic fields (Q2687373)

It is widely believed that the only number fields with class number \(1\) that are not Euclidean with respect to a suitable function are the imaginary quadratic number fields with discriminants \(-19\), \(-43\), \(-67\) and \(-163\). In this article, the authors use the method by \textit{M. Harper} and \textit{M. R. Murty} [Can. J. Math. 56, No. 1, 71--76 (2004; Zbl 1048.11080)] to show that all imaginary biquadratic number fields \({\mathbb Q}(\sqrt{m},\sqrt{n})\) and all imaginary cyclic quartic number fields with class number \(1\) are Euclidean; this is done by finding admissible primes for each of the finitely many fields.