# Irreducible Polynomials over Finite Fields

by burritoloco
Tags: fields, finite, irreducible, polynomials
 P: 85 Hi, yet another question regarding polynomials :). Just curious about this. Let f(x), g(x) be irreducible polynomials over the finite field GF(q) with coprime degrees n, m resp. Let $\alpha , \beta$ be roots of f(x), g(x) resp. Then the roots of f(x), g(x), are $\alpha^{q^i}, 0\leq i \leq n-1$, and $\beta^{q^j}, 0\leq j \leq m-1$. Question: What is the irreducible polynomial over GF(q) of degree nm with roots $\alpha^{q^i}\beta^{q^j}$ where $0\leq i \leq n-1$, and $0\leq j \leq m-1$. Can you define such polynomial explicitly in terms of just f(x) and g(x) without the roots appearing in the formula? Note: The last sentence/question is what really interests me as the following is the required polynomial (but defined in terms of the roots of f(x)) $$F(x) = \prod_{i=0}^{n-1}\alpha^{mq^i}g\left(\alpha^{-q^i}x\right)$$ Thank you!
 Emeritus Sci Advisor PF Gold P: 16,091 It depends somewhat on just what you mean by "formula" and "appearing in". I know that F can be expressed as a resultant. The formula you give is presumably comes from one of the methods of computing resultants. There exist other ways to compute resultants, such as as the determinant of a matrix, or a Euclidean algorithm-like method. You could always find a way to write down the system of equations that literally expresses that $\alpha \beta$ is a root of F(x). Then F would be the solution!