• Title/Summary/Keyword: cyclotomic function fields

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On the ring of integers of cyclotomic function fields

  • Bae, Sunghan;Hahn, Sang-Geun
    • Bulletin of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.153-163
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    • 1992
  • Carlitz module is used to study abelian extensions of K=$F_{q}$(T). In number theory every abelian etension of Q is contained in a cyclotomic field. Similarly every abelian extension of $F_{q}$(T) with some condition on .inf. is contained in a cyclotomic function field. Hence the study of cyclotomic function fields in analogy with cyclotomic fields is an important subject in number theory. Much are known in this direction such as ring of integers, class groups and units ([G], [G-R]). In this article we are concerned with the ring of integers in a cyclotomic function field. In [G], it is shown that the ring of integers is generated by a primitive root of the Carlitz module using the ramification theory and localization. Here we will give another proof, which is rather elementary and explicit, of this fact following the methods in [W].[W].

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THE p-PART OF DIVISOR CLASS NUMBERS FOR CYCLOTOMIC FUNCTION FIELDS

  • Daisuke Shiomi
    • Communications of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.715-723
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    • 2023
  • In this paper, we construct explicitly an infinite family of primes P with h±P ≡ 0 (mod qdeg P), where h±P are the plus and minus parts of the divisor class number of the P-th cyclotomic function field over 𝔽q(T). By using this result and Dirichlet's theorem, we give a condition of A, M ∈ 𝔽q[T] such that there are infinitely many primes P satisfying with h±P ≡ 0 (mod pe) and P ≡ A (mod M).

CYCLOTOMIC UNITS AND DIVISIBILITY OF THE CLASS NUMBER OF FUNCTION FIELDS

  • Ahn, Jae-Hyun;Jung, Hwan-Yup
    • Journal of the Korean Mathematical Society
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    • v.39 no.5
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    • pp.765-773
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    • 2002
  • Let $textsc{k}$$F_{q}$(T) be a rational function field. Let $\ell$ be a prime number with ($\ell$, q-1) = 1. Let K/$textsc{k}$ be an elmentary abelian $\ell$-extension which is contained in some cyclotomic function field. In this paper, we study the $\ell$-divisibility of ideal class number $h_{K}$ of K by using cyclotomic units.s.s.

ON THE RELATIVE ZETA FUNCTION IN FUNCTION FIELDS

  • Shiomi, Daisuke
    • Communications of the Korean Mathematical Society
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    • v.27 no.3
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    • pp.455-464
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    • 2012
  • In the previous paper [8], the author gave a determinant formula of relative zeta function for cyclotomic function fields. Our purpose of this paper is to extend this result for more general function fields. As an application of our formula, we will give determinant formulas of class numbers for constant field extensions.