Algebraic Extension help!

Question

I'm having severe difficultly proving this statement.

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\deg f$ and $\deg g$ are relatively prime. If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$.

This problem comes from "Contemporary Abstract Algebra" 7th edition by Joseph A. Gallian.

Answer 1

Let $a,b$ be any roots of $f(x),g(x)$ respectively taken from an extension field.

• Subexercise One: explain why $[F(a,b):F]=(\deg f)\cdot(\deg g)$ suffices to show that $g(x)$ is irreducible over $F(a)$ (and, symmetrically, $f(x)$ is irreducible over $F(b)$). Hint: consider the extension $F(a,b)$ over $F(a)$ and $b$'s minimal polynomial over $F(a)$ versus $g(x)$. Also be sure to use the transitive property $L/M/N\Rightarrow [L:N]=[L:M][M:N]$.
• Subexercise Two: actually show $[F(a,b):F]=(\deg f)(\deg g)$. Hint: show each degree on the right divides the index on the left and invoke coprimality, then argue the left hand side is bounded above by the right-hand side.