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Nonstandard Arithmetic of Polynomial Rings

Published online by Cambridge University Press:  22 January 2016

Masahiro Yasumoto
Masahiro Yasumoto*
Affiliation:
Department of Mathematics Faculty of Science Nagoya University, Chikusa-ku, Nagoya 464, Japan
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Let f(X, T1,…, Tm) be a polynomial over an algebraic number field k of finite degree. In his paper [2], T. Kojima proved

THEOREM. Let K = Q. if for every m integers t1, …, tm, there exists an r ∈ K such that f(r, t1, …, tm) =), then there exists a rational function g(T1,…,Tm) over Q such that

F(g(T1,…,Tm), T1,…,T)= 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 105 , March 1987 , pp. 33 - 37
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[ 1 ] Davenport, H., Lewis, D. J. and Schinzel, A., Polynomials of certain special types, Acta Arith., 9 (1964), 107–116.Google Scholar
[ 2 ] Kojima, T., Note on number theoretic properties of algebraic functions, Tohoku Math. J., 8 (1915), 24–37.Google Scholar
[ 3 ] Robinson, A. and Roquette, P., On the finiteness theorem of Siegel and Mahler concerning diophantine equations, J. Number Theory, 7 (1975), 121–176.Google Scholar
[ 4 ] Roquette, P., Nonstandard aspects of Hubert’s irreducible theorem, Lecture Notes in Math., 498, 231–274.CrossRefGoogle Scholar
[ 5 ] Schinzel, A., On Hilbert’s irreducible theorem, Ann. Polon. Math., 16 (1965), 333–340.CrossRefGoogle Scholar
[ 6 ] Schinzel, A., Selected topics on polynomials, Michigan 1982.Google Scholar
[ 7 ] Tung, S., On weak number theories, Thesis Illinois 1983.Google Scholar
[ 8 ] T. Skolem Einige Satze uber Polynome, Avhandlinger Norske Vid. Akad. Oslo, I Mat-Naturv. Kl. No. 4.Google Scholar