IRREDUCIBILITY OF POLYNOMIALS AND DIOPHANTINE EQUATIONS

Title & Authors
IRREDUCIBILITY OF POLYNOMIALS AND DIOPHANTINE EQUATIONS
Woo, Sung-Sik;

Abstract
In [3] we showed that a polynomial over a Noetherian ring is divisible by some other polynomial by looking at the matrix formed by the coefficients of the polynomials which we called the resultant matrix. In this paper, we consider the polynomials with coefficients in a field and divisibility of a polynomial by a polynomial with a certain degree is equivalent to the existence of common solution to a system of Diophantine equations. As an application we construct a family of irreducible quartics over $\small{\mathbb{Q}}$ which are not of Eisenstein type.
Keywords
irreducible polynomial;Diophantine equations;rational points of elliptic curves;
Language
English
Cited by
1.
CUBIC FORMULA AND CUBIC CURVES,;

대한수학회논문집, 2013. vol.28. 2, pp.209-224
1.
CUBIC FORMULA AND CUBIC CURVES, Communications of the Korean Mathematical Society, 2013, 28, 2, 209
References
1.
N. Bourbaki, Elements of Mathematics, Algebra I, Addison-Wesley, 1973

2.
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986

3.
S. S. Woo, Dividing polynomials using the resultant matrix, Comm. Algebra 35 (2007), no. 11, 3263-3272