• 충청수학회지
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• 제4권1호
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• pp.85-90
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• 1991

In this article we introduce the readers to the theory of Carlitz modules which are rank one Drinfeld modules. The main point is the striking similarities between cyclotomic number fields and Carlitz modules.

• 호남수학학술지
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• 제37권2호
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• pp.265-267
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• 2015

Let G be a cyclic group of prime power order. There is a natural embedding of $\mathbb{Z}$[G] into a product of rings of integers of cyclotomic fields. In this paper we determine the image of powers of the augmentation ideal.

• 대한수학회논문집
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• 제30권1호
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• pp.1-5
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• 2015

Let G be a cyclic group of prime power order. There is a natural embedding of $\mathbb{Z}[G]$ into a product of rings of integers of cyclotomic fields. In this paper the image of the embedding is determined, and we also compute the index of the image.

• 충청수학회지
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• 제20권4호
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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].

• Korean Journal of Mathematics
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• 제4권1호
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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 $m>n{\geq}0$.

• 대한전자공학회:학술대회논문집
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• 대한전자공학회 2006년도 하계종합학술대회
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• pp.1053-1054
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• 2006

In this paper we consider a family {$H_m$},m =1,2,..., of generalized Hadamard matrices of order $P^m$, where p is a prime number, and construct the corresponding family {$C^*_m$} of generalize p-ary Hadarmard codes which meet the Plotkin bound. Index terms: Cyclotomic fields, cocyclic matrices, Butson-Hadamard matrices, generalized Hadamard codes, decoding.

• 대한수학회보
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• 제20권1호
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• pp.31-36
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• 1983

Let R be a commutative ring with 1, and A a unitary commutative R-algebra. By a derivation module of A, we mean a pair (M, d), where M is an A-module and d: A.rarw.M and R-derivation, i.e., d is an R-linear mapping such that d(ab)=a)db)+b(da). A derivation module homomorphism f:(M,d).rarw.(N, .delta.) is an A-homomorphism f:M.rarw.N such that f.d=.delta.. A derivation module of A, (U, d), there exists a unique derivation module homomorphism f:(U, d).rarw.(M,.delta.). In fact, a universal derivation module of A exists in the category of derivation modules of A, and is unique up to unique derivation module isomorphisms [2, pp. 101]. When (U,d) is a universal derivation module of R-algebra A, the A-module U is denoted by U(A/R). For out convenience, U(A/R) will also be called a universal derivation module of A, and d the R-derivation corresponding to U(A/R).

• Korean Journal of Mathematics
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• 제18권2호
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• pp.161-166
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• 2010

For an abelian extension K of ${\mathbb{Q}}$, let $C_W(K)$ be the group of Washington units of K, and $C_S(K)$ the group of Sinnott units of K. A lot of results about $C_S(K)$ have been found while very few is known about $C_W(K)$. This is mainly because elements in $C_S(K)$ are more explicitly defined than those in $C_W(K)$. The aim of this paper is to find a basis of $C_W(K)$ and use it to compare $C_W(K)$ and $C_S(K)$ when K is a subfield of ${\mathbb{Q}}({\zeta}_{p^e})$, where p is a prime.

• Journal of Communications and Networks
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• 제13권1호
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• pp.12-16
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• 2011

In this paper, we construct two families $C^*_m$ and ${\~{C}}^*_m$ of ternary ($2^m$, $3^m$, $2^{m-1}$ ) and ($2^m$, $3^{m+1}$, $2^{m-1}$ ) codes, for m = 1, 2, 3, ${\cdots}$, derived from the corresponding families of modified ternary Jacket matrices. These codes are close to the Plotkin bound and have a very easy decoding procedure.