IRREDUCIBILITY OF POLYNOMIALS AND DIOPHANTINE EQUATIONS
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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 Q which are not of Eisenstein type.
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 Q which are not of Eisenstein type.
참고문헌 (3)
- N. Bourbaki, "Elements of Mathematics, Algebra I" (1973), Addison-Wesley
- J. H. Silverman, "The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106" (1986), Springer-Verlag
- S. S. Woo, "Dividing polynomials using the resultant matrix", Comm. Algebra, 35(11), pp.3263~3272 (2007)