• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.25-43
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• 2005

Let F be a finite field of prime power order q(odd) and the multiplicative order of q modulo $2^{n}\;(n>1)\;be\; {\phi}(2^{n})/2$. If n > 3, then q is odd number(prime or prime power) of the form $8m{\pm}3$. If q = 8m - 3, then the ring $R_{2^n} = F[x]/ < x^{2^n}-1 >$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length $2^{n}$ generated by these idempotents are completely described. If q = 8m + 3 then the expressions for the 2n - 1 primitive idempotents of $R_{2^n}$ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n-1 idempotents are also obtained. The case n = 2,3 is dealt separately.

• Communications of Mathematical Education
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• v.19 no.4 s.24
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• pp.599-620
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• 2005

There can be two different approaches to introducing Abstract Algebra to undergraduate students: One is to introduce group concept prior to ring concept, and the other is to do the other way around. Although the former is almost conventional, it is worth while to take the latter into consideration in the viewpoint that students are already familiar to rings of integers and polynomials. In this paper, we investigated 16 most commonly used Abstract Algebra undergraduate textbooks and found that 5 of them introduce ring theory prior to group theory while the rest do the other way around. In addition, we interviewed several undergraduate students who already have taken an Abstract Algebra course to look into which approach they prefer. Then we compare pros and cons of two approaches on the basis of the results of the interview and the historico-genetic principle of teaching and learning in Abstract Algebra and suggest that it certainly be one of alternatives to introduce ring theory before group theory in its standpoint.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.425-435
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• 2003

In this paper, a new kind of matrices, i.e., $level-{\kappa}$ II-circulant matrices is considered. Algorithms for computing minimal polynomial of this kind of matrices are presented by means of the algorithm for the Grobner basis of the ideal in the polynomial ring. Two algorithms for finding the inverses of such matrices are also presented based on the Buchberger's algorithm.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.799-814
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• 2009

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, using Shimura Reciprocity Law, we compute the Galois actions of some class invariants from the generalized Weber functions $\mathfrak{g}_0,\mathfrak{g}_1,\mathfrak{g}_2$ and $\mathfrak{g}_3$ over quadratic number fields with discriminant $D{\equiv}1$ (mod 12).

• Kyungpook Mathematical Journal
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• v.28 no.1
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• pp.39-44
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• 1988

In 1940, B. H. Neumann, working with a system more general than a near-field, proved that the additive group of such a system (and of a near-field) is commutative. The algebraic structure he used is known as a Neumann system (N-system). Here, the prime N-systems are classified and for each possible characteristic, examples of N-systems which are neither near-fields nor rings are given. It is also shown that a necessary condition for the set of all odd polynomials over GF(p) to be an N-system is that p is a Fermat prime.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.1
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• pp.9-16
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• 2014

Let $\mathbb{A}=\mathbb{F}_q[T]$ be a polynomial ring over $\mathbb{F}_q$. In this paper we determine an asymptotic mean value of quadratic Dirich-let L-functions L(s, ${\chi}_{{\gamma}D}$) at the central point s=$\frac{1}{2}$, where D runs over all monic square-free polynomials of even degree in $\mathbb{A}$ and ${\gamma}$ is a generator of $\mathbb{F}_q^*$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.699-709
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• 2016

Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.

• East Asian mathematical journal
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• v.36 no.5
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• pp.537-545
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• 2020

We investigate some polynomial invariants for virtual knots via virtualization moves. We define two types of polynomials W_G(t) and S²_G(t) from the definition of the index polynomial for virtual knots. And we show that they are invariants for virtual knots on the quotient ring Z[t^±1]/I where I is an ideal generated by t² - 1.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.751-755
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• 2008

Formanek([2]) proved that $M_n(K)$, the matrix algebra has a nontrivial central polynomial when char K = 0. Also Razmyslov([3]) showed the same result using the essential weak identity. In this article we explicitly compute Formanek's central polynomial for $M_2(\mathbb{C})$ and $M_3(\mathbb{C})$ and classify the coefficients of the central polynomial.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.101-129
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• 2018

From an analytical perspective, we introduce a sequence of Cartier operators that act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic p. In this work, we discover the binomial inversion formula between Hasse derivatives and Cartier operators, implying that Cartier operators can play a prominent role in various objects of study in function field arithmetic, as a suitable substitute for higher derivatives. For an applicable object, the Wronskian criteria associated with Cartier operators are introduced. These results stem from a careful study of two types of Cartier operators on the power series ring ${\mathbf{F}}_q$[[T]] in one variable T over a finite field ${\mathbf{F}}_q$ of q elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous ${\mathbf{F}}_q$-linear functions on ${\mathbf{F}}_q$[[T]]. According to the digit principle, every continuous function on ${\mathbf{F}}_q$[[T]] is uniquely written in terms of a q-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of the two cases. Moreover, the p-adic analogues of Cartier operators are discussed as orthonormal bases for the space of continuous functions on ${\mathbf{Z}}_p$.