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Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
Introduction and the statement of results.................................................................5
Part I. Reducibility over function fields...................................................................14
1. Auxiliary results from the theory of algebraic functions....................................14
2. Determination of the range of Tables 1 and 2 (Lemmas 3-27).........................15
3. Determination of the content of Table 1 (Lemmas 28-40)................................38
4. Determination of the content of Table 2 (Lemmas 41-48)................................46
5. Proof of Theorems 1, 2 and 3..........................................................................55
6. Proof of Theorems 4 and 5..............................................................................58
Part II. Reducibility over algebraic number fields and, in particular, over ℚ............61
7. Proof of Theorem 6 and of the subsequent remarks.......................................61
8. Deduction of Consequences 1-3 from Conjecture...........................................65
9. Proof of Theorems 7 and 8..............................................................................66
10. Proof of Theorem 9 and of Corollary 1..........................................................68
11. Proof of Theorem 10 and of Corollary 2........................................................75
References............................................................................................................82
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
329
Liczba stron
83
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXXIX
Daty
wydano
1993
otrzymano
1993-04-09
poprawiono
1993-06-02
Twórcy
autor
- Institute of Mathematics, Polish Academy of Sciences, P.O. Box 137, Śniadeckich 8, 00-950 Warszawa, Poland
Bibliografia
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- [3] A. Capelli, Sulla riduttibilità delle equazioni algebriche, Nota prima, Rend. Accad. Sci. Fis. Mat. Soc. Napoli (3) 3 (1897), 243-252.
- [4] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193-291.
- [5] C. Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, New York, 1951.
- [6] A. Choudhry and A. Schinzel, On the number of terms in the irreducible factors of a polynomial over ℚ, Glasgow Math. J. 34 (1992), 11-15.
- [7] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401.
- [8] M. Eichler, Einführung in die Theorie der algebraischen Zahlen und Funktionen, Birkhäuser, Basel-Stuttgart, 1963.
- [9] M. Fried, On the Diophantine equation f(y)-x = 0, Acta Arith. 19 (1971), 79-87.
- [10] M. Fried, Exposition on an arithmetic-group theoretic connection via Riemann's existence theorem, in: Proc. Sympos. Pure Math. 37, Amer. Math. Soc., 1980, 571-602.
- [11] M. Fried and A. Schinzel, Reducibility of quadrinomials, Acta Arith. 21 (1972), 153-171.
- [12] K. Győry and A. Schinzel, On a conjecture of Posner and Ramsey, J. Number Theory, to appear.
- [13] E. Maillet, Détermination des points entiers des courbes algébriques unicursales à coefficients entiers, C. R. Acad. Sci. Paris 168 (1919), 217-220.
- [14] H. Mann, On linear relations between roots of unity, Mathematika 12 (1965), 107-117.
- [15] T. Nagell, Sur la réductibilité des trinômes, in: Comptes rendus du 8. congrès des mathématiciens scandinaves, Stockholm, 1934, 273-275.
- [16] T. Nagell, Sur la classification des cubiques planes du premier genre par des transformations birationnelles dans un domaine de rationalité quelconque, Nova Acta Regiae Soc. Sci. Upsaliensis (4) 12 (1941), No. 8.
- [17] T. Nagell, Contributions à la théorie des corps et des polynômes cyclotomiques, Ark. Mat. 5 (1963), 153-192.
- [17A] Numerical tables on elliptic curves, in: Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer, Berlin, 1975.
- [18] L. Rédei, Algebra. Erster Teil, Akad. Verlagsges. Geest & Portig K.-G., Leipzig, 1959.
- [19] H. Reichardt, Über die Diophantische Gleichung ax⁴ + bx²y² + cy⁴ = ez², Math. Ann. 117 (1940), 235-276.
- [20] P. Ribenboim, On the factorization of $x^n - Bx - A$, Enseign. Math. 37 (1991), 191-200.
- [21] A. Schinzel, Some unsolved problems on polynomials, in: Neki nerešeni problemi u matematici, Mat. Bibl. 25, Zavod Izd. Udžb., Belgrade, 1963, 63-70.
- [22] A. Schinzel, Reducibility of lacunary polynomials I, Acta Arith. 16 (1969), 123-159.
- [23] A. Schinzel, Reducibility of polynomials, in: Computers in Number Theory, Academic Press, London, 1971, 73-75.
- [24] A. Schinzel, Primitive divisors of the expression $A^n - B^n$ in algebraic number fields, J. Reine Angew. Math. 268/269 (1974), 27-33.
- [25] A. Schinzel, On linear dependence of roots, Acta Arith. 28 (1975), 161-175.
- [26] A. Schinzel, Selected Topics on Polynomials, University of Michigan Press, Ann Arbor, 1982.
- [27] E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand. 4 (1956), 287-302.
- [28] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuß. Akad. Wiss., Phys.-math. Kl. 1929 Nr 1.
- [29] C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169-175.
- [30] N. Tschebotaröw und H. Schwerdtfeger, Grundzüge der Galois'schen Theorie, Noordhoff, Groningen-Djakarta, 1950.
- [31] G. Turnwald, On Schur's conjecture, J. Austral. Math. Soc. Ser. A, to appear.
- [32] K. Th. Vahlen, Über reductible Binome, Acta Math. 19 (1895), 195-198.
Języki publikacji
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Uwagi
1991 Mathematics Subject Classification: 12E05, 12E10.
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