• Title/Summary/Keyword: cyclotomic units

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HIGHER CYCLOTOMIC UNITS FOR MOTIVIC COHOMOLOGY

  • Myung, Sung
    • Korean Journal of Mathematics
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    • v.21 no.3
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    • pp.331-344
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    • 2013
  • In the present article, we describe specific elements in a motivic cohomology group $H^1_{\mathcal{M}}(Spec\mathbb{Q}({\zeta}_l),\;\mathbb{Z}(2))$ of cyclotomic fields, which generate a subgroup of finite index for an odd prime $l$. As $H^1_{\mathcal{M}}(Spec\mathbb{Q}({\zeta}_l),\;\mathbb{Z}(1))$ is identified with the group of units in the ring of integers in $\mathbb{Q}({\zeta}_l)$ and cyclotomic units generate a subgroup of finite index, these elements play similar roles in the motivic cohomology group.

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.

GENERATORS OF COHOMOLOGY GROUPS OF CYCLOTOMIC UNITS

  • Kim, Jae Moon;Oh, Seung Ik
    • Korean Journal of Mathematics
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    • v.5 no.1
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    • pp.61-74
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    • 1997
  • Let $d$ be a positive integer with $d\not{\equiv}2$ mod 4, and let $K=\mathbb{Q}({\zeta}_{pd})$ for S an odd prime $p$ such that $p{\equiv}1$ mod $d$. Let $K_{\infty}={\bigcup}_{n{\geq}0}K_n$ be the cyclotomic $\mathbb{Z}_p$-extension of $K=K_0$. In this paper, explicit generators for the Tate cohomology group $\hat{H}^{-1}$($G_{m,n}$ are given when $d=qr$ is a product of two distinct primes, where $G_{m,n}$ is the Galois group Gal($K_m/K_n$) and $C_m$ is the group of cyclotomic units of $K_m$. This generalizes earlier results when $d=q$ is a prime.

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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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ANTI-CYCLOTOMIC EXTENSION AND HILBERT CLASS FIELD

  • Oh, Jangheon
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.1
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    • pp.91-95
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    • 2012
  • In this paper, we show how to construct the first layer $k^{\alpha}_{1}$ of anti-cyclotomic ${\mathbb{{Z}}}_{3}$-extension of imaginary quadratic fields $k(=\;{\mathbb{{Q}}}(\sqrt{-d}))$ when the Sylow subgroup of class group of k is 3-elementary, and give an example. This example is different from the one we obtained before in the sense that when we write $k^{\alpha}_{1}\;=\;k({\eta}),{\eta}$ is obtained from non-units of ${\mathbb{{Q}}}({\sqrt{3d}})$.

CIRCULAR UNITS IN A BICYCLIC FUNCTION FIELD

  • Ahn, Jaehyun;Jung, Hwanyup
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.1
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    • pp.61-69
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    • 2008
  • For a real subextension of some cyclotomic function field with a non-cyclic Galois group order $l^2$, l being a prime different from the characteristic of function field, we compute the index of the Sinnott group of circular units.

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A NOTE ON CYCLOTOMIC UNITS IN FUNCTION FIELDS

  • Jung, Hwanyup
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.4
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    • pp.433-438
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    • 2007
  • Let $\mathbb{A}=\mathbb{F}_q[T]$ and $k=\mathbb{F}_q(T)$. Assume q is odd, and fix a prime divisor ${\ell}$ of q - 1. Let P be a monic irreducible polynomial in A whose degree d is divisible by ${\ell}$. In this paper we define a subgroup $\tilde{C}_F$ of $\mathcal{O}^*_F$ which is generated by $\mathbb{F}^*_q$ and $\{{\eta}^{{\tau}^i}:0{\leq}i{\leq}{\ell}-1\}$ in $F=k(\sqrt[{\ell}]{P})$ and calculate the unit-index $[\mathcal{O}^*_F:\tilde{C}_F]={\ell}^{\ell-2}h(\mathcal{O}_F)$. This is a generalization of [3, Theorem 16.15].

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ON THE IDEAL CLASS GROUPS OF REAL ABELIAN FIELDS

  • Kim, Jae Moon
    • Korean Journal of Mathematics
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    • v.4 no.1
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    • pp.45-49
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    • 1996
  • Let $F_0$ be the maximal real subfield of $\mathbb{Q}({\zeta}_q+{\zeta}_q^{-1})$ and $F_{\infty}={\cup}_{n{\geq}0}F_n$ be its basic $\mathbb{Z}_p$-extension. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $F_n$. The aim of this paper is to examine the injectivity of the natural $mapA_n{\rightarrow}A_m$ induced by the inclusion $F_n{\rightarrow}F_m$ when $m>n{\geq}0$. By using cyclotomic units of $F_n$ and by applying cohomology theory, one gets the following result: If $p$ does not divide the order of $A_1$, then $A_n{\rightarrow}A_m$ is injective for all $m>n{\geq}0$.

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