• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.23-26
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• 2014

Let E be a graded central division algebra (GCDA) over a grade field R. Let S be an unramified graded field extension of R. We describe the grading on the underlying GCDA E' of $E{\otimes}_RS$ which is analogous to the valuation on a tame division algebra over Henselian valued field.

• Honam Mathematical Journal
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• v.42 no.2
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• pp.213-218
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• 2020

Assume that K is a number field and K_ur is the maximal unramified extension of it. When Gal(K_ur/K) is an infinite group. It is known that Gal(K_ur/K) is generated by finitely many Frobenius classes of Gal(K_ur/K) by Y. Ihara. In this paper, we will give the explicit number of Frobenius classes which generate whole group Gal(K_ur/K).

• Communications of the Korean Mathematical Society
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• v.24 no.1
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• pp.1-4
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• 2009

In this paper we explicitly compute a Minkowski unit of a real abelian field and give a criterion when the first layer of anti-cyclotomic ${\mathbb{Z}}_3$-extension of an imaginary quadratic field is unramified everywhere.

• Journal of Integrative Natural Science
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• v.14 no.4
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• pp.159-161
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• 2021

In this article, we will study which prime numbers are represented by the principal form. p = x² + ny². We will also give some results related to the form of primes of the form x² + axy + y².

• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1249-1268
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• 2011

For imaginary quadratic number fields F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$, where ${\varepsilon}{\in}${-1,-2} and distinct primes $p_i{\equiv}1$ mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of $p_i$ modulo $2^n$, the quartic residue symbol $(\frac{p_1}{p_2})4$ and binary quadratic forms such as $p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$, where $h+(2p_1)$ is the narrow class number of $\mathbb{Q}(\sqrt{2p_1})$. The results are also very useful for numerical computations.