2025/08/11 by Md. Imdadul Islam, Islam, Md. Imdadul, Debopam Chakraborty +3 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2508.07894
openalex publication_date 2025/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Pólya group Po(K) of an algebraic number field K is the subgroup of the ideal class group ClK generated by the ideal classes of the products of prime ideals of the same norm. If Po(K) is trivial, then the number field K is said to be a Pólya field. In this article, we furnish three families ℚ(√(p),√(qrs)), ℚ(√(2p),√(qrs)) and ℚ(√(2p),√(2qrs)) of bi-quadratic Pólya fields K involving prime numbers p,q,r and s that satisfy certain quadratic residue conditions. It is worthwhile to note that in each of the fields, exactly five primes ramify in K/ℚ and this is the maximum possible number of ramified primes in a Pólya field over ℚ. Towards the end of the paper, we discuss about large Pólya groups of the compositums of Shank's cubic fields and Lehmer's quintic fields and prove that there are infinitely many such fields with index 1.