"Suppose that f(x) and g(x) are irreducible over F and that deg f(x) and deg g(x) are relatively prime. Suppose $a$ is a zero of f(x) in some extension field of F. Show that g(x) is irreducible over F(a)."
This is from Gallian's Abstract Algebra textbook. The one problem I have so far is how do we know $a$ is also not a zero of g(x)? The problem does not assume that $a$ is only just a zero of f(x) but not g(x). If it is a zero of g(x), then it has a linear factor over F(a) and it is reducible.
Another point: most of the answers to this problem starts with "Let b be a root of g(x) in some extension field of F(a)"... But doesn't that already assume that that g(x) cannot be reducible over F(a) which is what we are trying to prove? Why not say "Let b be a root of g(x) in some extension field of F(a) and b is not a root of f(x)" Because that way, we eliminate the possibility that F(a,b)=F(a).