Solutions to final review sheet
... Solution. (i) An irreducible element is a non-zero, non-unit x such that if x = yz for y, z ∈ R, either
y or z is a unit.
(ii) (a) (x + 1)2 ; (b) (x + 1)(x2 + x + 1); (c) x3 + x + 1 (irreducible as no roots); (d) x3 + x2 + 1
(irreducible as no roots); (e) (x + 1)3 .
(iii) (a) NO its a unit; (b) We k ...
Resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called eliminant.The resultant is widely used in number theory, either directly or through the discriminant, which is essentially the resultant of a polynomial and its derivative. The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of computer algebra, and is a built-in function of most computer algebra systems. It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation.The resultant of n homogeneous polynomials in n variables or multivariate resultant, sometimes called Macaulay's resultant, is a generalization of the usual resultant introduced by Macaulay. It is, with Gröbner bases, one of the main tools of effective elimination theory (elimination theory on computers).
