So basically I have the following: Let $F$ be a field, and let $f(t),g(t)\in F[t]$. Let $u=\frac{f}{g}$ be such that $u\in F(t)\backslash F$, and furthermore assume that $f$ and $g$ are relatively prime. Prove that $[F(t):F(u)]$ is finite and it is equal to $\max\{\deg(f),\deg(g)\}$.
The first part is easy since we can define $h(x)\in F(u)[x]$ by $h(x)=u\cdot g(x)-f(x)$. This is a polynomial with coefficients in $F(u)$ since $g$ and $f$ have coefficients in $F$. Note that $h(t)=0$, hence we see that $t$ is algebraic:\Rightarrow [F(t):F(u)]<\inftyNow in order to calculate the degree I have to either find the minimal polynomial of $t$ over $F(u)$, or find a $F(u)$-basis for $F(t)$ and count its dimension. I believe that $h$ might be the minimal polynomial but I cannot see why it is irreducible. Also, I tried looking for a basis, but I could not come up with one.
This is not homework, so you can go ahead and explain as thoroughly as you want.
Thanks