On reducible trinomials

Schinzel, Andrzej

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Tytuł książki

On reducible trinomials

Autorzy

Andrzej Schinzel

Seria

Rozprawy Matematyczne tom/nr w serii: 329 wydano: 1993

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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:

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Seria

Rozprawy Matematyczne tom/nr w serii: 329

Liczba stron

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCCXXIX

Daty

wydano

1993

otrzymano

1993-04-09

poprawiono

1993-06-02

Twórcy

autor

Andrzej Schinzel

Institute of Mathematics, Polish Academy of Sciences, P.O. Box 137, Śniadeckich 8, 00-950 Warszawa, Poland, schinzel@impan.impan.gov.pl

Bibliografia

[1] A. Bremner, On reducibility of trinomials, Glasgow Math. J. 27 (1981), 155-156.
[2] A. Bremner, On trinomials of type $x^n + Ax^m + 1$, Math. Scand. 49 (1981), 145-155.
[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

Uwagi

1991 Mathematics Subject Classification: 12E05, 12E10.

Identyfikator YADDA

bwmeta1.element.zamlynska-aa3b2a30-9a73-456e-be32-024a741a26ec

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ISSN

0012-3862

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DML-PL

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