• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.509-520
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• 2003

Let p be a prime number, $Q_{p}$ the field of p-adic numbers, K a finite extension of $Q_{p}$, $\bar{K}}$ a fixed algebraic closure of K and $C_{p}$ the completion of K with respect to the p-adic valuation. Let E be a closed subfield of $C_{p}$, containing K. Given elements $t_1$...,$t_{r}$ $\in$ E for which the field K($t_1$...,$t_{r}$) is dense in E, we construct integral bases of E over K.

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.537-544
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• 2018

There is a standard construction of lifting cyclic codes over the prime finite field ${\mathbb{Z}}_p$ to the rings ${\mathbb{Z}}_{p^e}$ and to the ring of p-adic integers. We generalize this construction for arbitrary finite fields. This will naturally enable us to lift codes over finite fields ${\mathbb{F}}_{p^r}$ to codes over Galois rings GR($p^e$, r). We give concrete examples with all of the lifts.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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• pp.1-14
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• 2008

In this paper, we prove the Hyers-Ulam-Rassias stability of the Cauchy functional equation f(x+y) = f(x)+f(y) and of the Jensen functional equation $2f(\frac{x+y}{2})=f(x)+f(y)$ over the p-adic field ${\mathbb{Q}}_p$. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.3
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• pp.245-261
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• 2006

In this paper, we prove the generalized Hyers-Ulam stability of ring homomorphisms over the p-adic field $\mathbb{Q}_p$ associated with the Cauchy functional equation f(x+y) = f(x)+f(y) and the Cauchy-Jensen functional equation $2f(\frac{x+y}{2}+z)=f(x)+f(y)+2f(z)$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1463-1481
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• 2021

Divisibility properties of harmonic numbers by a prime number p have been a recurrent topic. However, finding the exact p-adic orders of them is not easy. Using class numbers of number fields and Bernoulli numbers, we compute the exact p-adic orders of harmonic type sums. Moreover, we obtain an asymptotic formula for generalized harmonic numbers whose p-adic orders are exactly one.

• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.595-606
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• 2001

Using p-adic class number formula, we derive a congru-ence relation for class numbers of the simplest cubic fields which can be considered as a cubic analogue of Ankeny-Artin-Chowlas theo-rem, Furthermore, we give an elementary proof for an upper bound for the class numbers of the simplest cubic fields.