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What should be assumed about the integral polynomials f1(x),... ,fk(,x) in order that the solvability of the congruence f1(x)f2(x) ... fk(x) is equivalent to0 (mod p) for sufficiently large primes p implies the solvability of the equation f1(x)f2(x) ... fk(x) = 0 in integers x? We provide some explicit characterizations for the cases when fj (x) are binomials or have cyclic splitting fields.
Wydawca
Rocznik
Tom
Strony
123--132
Opis fizyczny
Bibliogr. 10 poz.
Twórcy
autor
- Institute of Mathematics, Polish Academy of Sciences, P.O. Box 21, 00-956 Warszawa, Poland, skalba@impan.gov.pl
Bibliografia
- [1] N. G. Chebotarev, Der Hilbertsche Satz, Visti VUAN 1923, 3-7; Russian translation: Collected Works, Vol. 1, Moscow-Leningrad, 1949, Vol. 1, 14-17.
- [2] M. Filaseta and D. R. Richman, Sets which contain a quadratic residue mod p for almost all p, Math. J. Okayama Univ. 31 (1989), 1-8.
- [3] M. Fried, Arithmetical properties of value sets of polynomials, Acta Arith. 15 (1969), 91-115.
- [4] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN, Warszawa, 1990.
- [5] D. R. Richman, On q-th power residues, unpublished manuscript, 1987.
- [6] A. Schinzel, Abelian binomials, power residues and exponential congruences, Acta Arith. 32 (1977), 245-274; Addendum and corrigendum, 36 (1980), 101-104.
- [7] A. Schinzel and M. Skałba, On power residues, ibid. 108 (2003), 77-94.
- [8] M. Skałba, Power residue problem on elliptic curves, Manuscripta Math. 114 (2004), 37-43.
- [9] B. L. van der Waerden, Noch eine Bemerkung zu der Arbeit "Zur Arithmetik der Polynome" von U. Wegner in Math. Ann. 105, 628-631, Math. Ann. 109 (1934), 679-680.
- [10] J. Wójcik, Contributions to the theory of Kummer extensions, Acta Arith. 40 (1982), 155-174.
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Bibliografia
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bwmeta1.element.baztech-article-BAT5-0004-0014