2019/10/11 by Su Hu, Yan Li, Hu, Su +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1910.06252
openalex publication_date 2019/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields K. Comparing with the previous works, our methods yield a new larger family of real biquadratic fields K having Euclidean ideals; and the conditions for our family seem to be more efficient for the computations. Moreover, the previous approaches mainly focus on the case if hK=2, while the present approach can also deal with the general case when hK=2t (t≥1), where hK denotes the ideal class number of K. In particular, if hK≥ 4, it shows that H(K), the Hilbert class field of K, is always non-abelian over ℚ for the family of K given in this paper having Euclidean ideals, whereas the previous approaches always requires that H(K) is abelian over ℚ explicitly or implicitly. Finally, some open questions have also been listed for further research.