• 대한수학회논문집
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• 제27권3호
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• pp.449-454
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• 2012

In this paper, we find the minimal polynomial of a primitive root of unity over cyclotomic fields. From this, we factorize cyclotomic polynomials over cyclotomic fields and investigate the coefficients of ${\Phi}_{3n}(x)$ when 3∤$n$.

• 충청수학회지
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• 제29권1호
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• pp.117-121
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• 2016

In this paper we give a determinant formula for the ratio of relative congruence zeta functions of cyclotomic function fields.

• 대한수학회논문집
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• 제20권2호
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• pp.259-273
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• 2005

In this paper we determine all subfields with genus one of cyclotomic function fields over rational function fields explicitly.

• 충청수학회지
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• 제28권1호
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• pp.83-88
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• 2015

In this paper we give a determinant formula for the relative congruent zeta function in cyclotomic function fields.

• 대한수학회논문집
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• 제22권2호
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• pp.163-171
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• 2007

In this paper, we determine all subfields of cyclotomic function fields with divisor class number two. We also give the generators of such fields explicitly.

• 대한수학회보
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• 제29권1호
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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].

• 대한수학회지
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• 제56권2호
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• pp.559-578
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• 2019

We list all subfields of cyclotomic function fields over rational function fields with class number three. We also determine all the imaginary abelian extensions with relative class number three, explicitly.

• Korean Journal of Mathematics
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• 제21권3호
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• pp.265-270
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• 2013

In this paper, using a theorem of Brink for prime decomposition of the anti-cyclotomic extension, we explicitly construct the first layer of the anti-cyclotomic $\mathbb{Z}_3$-extension of imaginary quadratic fields.

• 대한수학회논문집
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• 제38권3호
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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 q^{deg P}), where h^±_P are the plus and minus parts of the divisor class number of the P-th cyclotomic function field over 𝔽_q(T). By using this result and Dirichlet's theorem, we give a condition of A, M ∈ 𝔽_q[T] such that there are infinitely many primes P satisfying with h^±_P ≡ 0 (mod p^e) and P ≡ A (mod M).

• 충청수학회지
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• 제24권3호
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• pp.595-599
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• 2011

We prove that for any positive integer $g{\geq}3$, there are ${\gg}q^{\frac{l}{2g}}$ real cyclotomic function fields whose conductor has degree ${\leq}l$ and ideal class number is divisible by $\frac{g}{gcd(2,g)}$.