Chattopadhyay, Jaitra; Muthukrishnan, Subramani Biquadratic fields having a non-principal Euclidean ideal class. (English) Zbl 1482.11147 J. Number Theory 204, 99-112 (2019). Extending work of C. Hsu [Int. J. Number Theory 12, No. 4, 1123–1136 (2016; Zbl 1415.11159)], the authors prove the biquadratic number fields \(K = \mathbb{Q}(\sqrt{q}, \sqrt{kr})\), where \(k \equiv r \equiv 1 \bmod 4\) and \(q = 2\) or \(q \equiv 3 \bmod 4\) are primes, have a Euclidean ideal class if \(K\) has class number \(2\). We remark that the construction of the Hilbert class field of \(K\) would have followed painlessly from genus theory applied to the quadratic number field \(\mathbb{Q}(\sqrt{qkr})\). Reviewer: Franz Lemmermeyer (Jagstzell) Cited in 3 Documents MSC: 11R29 Class numbers, class groups, discriminants 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11R16 Cubic and quartic extensions Keywords:Euclidean algorithm for number fields; Euclidean ideal class; quartic field; cyclic class group Citations:Zbl 1415.11159 PDFBibTeX XMLCite \textit{J. Chattopadhyay} and \textit{S. Muthukrishnan}, J. Number Theory 204, 99--112 (2019; Zbl 1482.11147) Full Text: DOI arXiv References: [1] Cox, D. A., Primes of the Form \(x^2 + n y^2\): Fermat, Class Field Theory and Complex Multiplication (1989), Wiley: Wiley New York · Zbl 0701.11001 [2] Graves, H., \(Q(\sqrt{2}, \sqrt{35})\) has a non-principal Euclidean ideal, Int. J. Number Theory, 7, 2269-2271 (2011) · Zbl 1256.11059 [3] Graves, H., Growth results and Euclidean ideals, J. Number Theory, 133, 2756-2769 (2013) · Zbl 1290.11140 [4] Graves, H.; Murty, M. Ram, A family of number fields with unit rank at least 4 that has Euclidean ideals, Proc. Amer. Math. Soc., 141, 2979-2990 (2013) · Zbl 1329.11115 [5] Harper, M.; Murty, M. Ram, Euclidean rings of algebraic integers, Canad. J. Math., 56, 71-76 (2004) · Zbl 1048.11080 [6] Hsu, C., Two classes of number fields with a non-principal Euclidean ideal, Int. J. Number Theory, 12, 1123-1136 (2016) · Zbl 1415.11159 [7] Lenstra, H. K., Euclidean ideal classes, Astérisque, 61, 121-131 (1979) · Zbl 0401.12005 [8] Marcus, D. A., Number Fields (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0383.12001 [9] Narkiewicz, W., Units in residue classes, Arch. Math., 51, 238-241 (1988) · Zbl 0641.12004 [10] Pollack, P., The least prime quadratic nonresidue in a prescribed residue class mod 4, J. Number Theory, 187, 403-414 (2018) · Zbl 1430.11006 [11] Weinberger, P. J., On Euclidean rings of algebraic integers, (Analytic Number Theory. Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, vol. 24 (1973), American Mathematical Society: American Mathematical Society Providence, RI), 321-332 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.