## Abstract

For certain two cases of splitting behaviors of the prime at infinity with unit rank r, given positive integers m,n, we construct infinitely many global function fields K such that the ideal class group of K of degree m over F(T) has n-rank at least m - r - 1 and the prime at infinity splits in K as given, where F denotes a finite field and T a transcendental element over F. In detail, for positive integers m, n and r with 0 ≤ r ≤ m - 1 and a given signature (e_{i},f_{i}), 1 ≤ i ≤ r + 1, such that Σ^{r+1}_{i=1} e_{i}f_{i} = m, in the following two cases where ei is arbitrary and f_{i} = 1 for each i, or e_{i} = 1 and f_{i}'s are the same for each i, we construct infinitely many global function fields K of degree m over F(T) such that the ideal class group of K contains a subgroup isomorphic to (ℤ/nℤ) ^{m-r-1}and the prime at infinity p;_{∞} splits into r + 1 primes β _{1}, β_{2}, • • •, P _{r+1} in K with e(β_{i}/p;_{∞}) = e _{i} and f(β_{i}/p;_{infin;}) = f_{i} for 1 ≤ i ≤ r + 1 (so, K is of unit rank r).

Original language | English |
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Pages (from-to) | 415-424 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2009 |

## Keywords

- Class group
- Class number
- Imaginary function field
- Rank of class group