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

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 ℓⁿ-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 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 K_i 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 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.