Infinite families of cyclotomic function fields with any prescribed class group rank

• Published in: Journal of pure and applied algebra 2021-09, Vol.225 (9), p.106658, Article 106658
• Main Authors: Yoo, Jinjoo, Lee, Yoonjin
• Format: Article
• Language: English
• Subjects: Class group rank; Cyclotomic function field; Ideal class group; Kummer extension; Maximal real subfield
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/j.jpaa.2020.106658

We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k=Fq(T) whose class groups can have arbitrarily largeℓn-rank, where Fq is the finite field of prime power order q. We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields k(Λ)+ of cyclotomic function fields k(Λ) whose ideal class groups have arbitrary ℓn-rank for n = 1, 2, and 3, where ℓ is a prime divisor of q−1. We also obtain a tower of cyclotomic function fields Ki whose maximal real subfields have ideal class groups of ℓn-ranks getting increased as the number of the finite places of k which are ramified in Ki get increased for i≥1. Our main idea is to use the Kummer extensions over k which are subfields of k(Λ)+, where the infinite prime ∞ of k splits completely. In fact, we construct the maximal real subfields k(Λ)+ of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k. We demonstrate our results by presenting some examples calculated by MAGMA at the end.