burritoloco

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 [itex]\alpha , \beta[/itex] be roots of f(x), g(x) resp. Then the roots of f(x), g(x), are [itex]\alpha^{q^i}, 0\leq i \leq n-1[/itex], and [itex]\beta^{q^j}, 0\leq j \leq m-1[/itex].

Question: What is the irreducible polynomial over GF(q) of degree nm with roots [itex]\alpha^{q^i}\beta^{q^j}[/itex] where [itex]0\leq i \leq n-1[/itex], and [itex]0\leq j \leq m-1[/itex]. 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))

[tex]F(x) = \prod_{i=0}^{n-1}\alpha^{mq^i}g\left(\alpha^{-q^i}x\right)[/tex]

Thank you!