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

Abstract

We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k = F q ( T ) whose class groups can have arbitrarily large ℓ n -rank, where F q 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 ℓ n -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 K i whose maximal real subfields have ideal class groups of ℓ n -ranks getting increased as the number of the finite places of k which are ramified in K i 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.