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

Jinjoo Yoo, Yoonjin Lee

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

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 large ℓn-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.

Original languageEnglish
Article number106658
JournalJournal of Pure and Applied Algebra
Volume225
Issue number9
DOIs
StatePublished - Sep 2021

Keywords

  • Class group rank
  • Cyclotomic function field
  • Ideal class group
  • Kummer extension
  • Maximal real subfield

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