## Abstract

We study the Galois module structure of the class groups of the Artin-Schreier extensions K over k of extension degree p, where k := Fq (T) is the rational function field and p is a prime number. The structure of the p-part clK ( p) of the ideal class group of K as a finite G-module is determined by the invariant γn, where G := Gal(K/k) = (σ) is the Galois group of K over k, and γn = dimFp (clK (p)^{(σ-1)n-1}/clK ( p) ^{(σ-1)n)}. We find infinite families of the Artin-Schreier extensions over k whose ideal class groups have guaranteed prescribed γn-rank for 1 ≤ n ≤ 3. We find an algorithm for computing γ3-rank of clK (p). Using this algorithm, for a given integer t ≥ 2, we get infinite families of the Artin-Schreier extensions over k whose γ1-rank is t, γ2-rank is t-1, and γ3-rank is t-2. In particular, in the case where p = 2, for a given positive integer t ≥ 2, we obtain an infinite family of the Artin-Schreier quadratic extensions over k whose 2-class group rank (respectively, 22-class group rank, 23-class group rank) is exactly t (respectively, t-1, t-2). Furthermore, we also obtain a similar result on the 2n-ranks of the divisor class groups of the Artin-Schreier quadratic extensions over k.

Original language | English |
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Journal | Canadian Journal of Mathematics |

DOIs | |

State | Accepted/In press - 2023 |

### Bibliographical note

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## Keywords

- Artin-Schreier extension
- Galois module.
- class group
- function field
- ideal class group