## 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.

Original language | English |
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Article number | 106658 |

Journal | Journal of Pure and Applied Algebra |

Volume | 225 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2021 |

### Bibliographical note

Publisher Copyright:© 2020 Elsevier B.V.

## Keywords

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