• Title, Summary, Keyword: Abelian function field

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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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THAINE'S THEOREM IN FUNCTION FIELD

  • Jung, Hwanyup
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.1
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    • pp.17-23
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    • 2009
  • Let F be a finite real abelian extension of a global function field k with G = Gal(F/k). Assume that F is an extension field of the Hilbert class field $K_e$ of k and is contained in a cyclotomic function field $K_n$. Let $\ell$ be any prime number not dividing $ph_k{\mid}G{\mid}$. In this paper, we show that if $\theta{\in}\mathbb{Z}[G]$ annihilates the Sylow $\ell$-subgroup of ${\mathcal{O}}^{\times}_F/{\mathcal{C}}_F$, then (q-1)$\theta$ annihilates the Sylow $\ell$-subgroup of ${\mathcal{Cl}}_F$.

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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 RELATIVE-INVARIANT CIRCULAR UNITS IN FUNCTION FIELDS

  • JUNG, HWANYUP
    • Honam Mathematical Journal
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    • v.27 no.3
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    • pp.389-397
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    • 2005
  • Let K be an absolutely real abelian number field with $G=Gal(K/{\mathbb{Q}})$. Let E be a subfield of K and ${\Delta}=Gal(K/E)$. Let $C_K$ and $C_E$ be the group of circular units of K and E respectively. In [G], Greither has shown that if G is cyclic then $C_K^{\Delta}=C_E$. In this paper we show that the same result holds in function field case.

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