I am trying to understand the solution to the following exercise from Milne's notes on Galois theory (available here):
At the end of the notes, the author presents the following solution to the problem:
Although I understand the argument, it is not immediately clear to me that this contradiction implies $F(X^2)\cap F(X^2-X)=F$. What immediately follows is that there are no polynomials of degree $m>0$ in this intersection. However, $E=F(X^2)\cap F(X^2-X)$ is a field of rational functions, so, in principle, it is not clear why there couldn't be non-constant rational functions $p(X)/q(X)$ in this field, even if there are no polynomials. More precisely, I can formulate my question as follows:
- How to show that, if there exists $p(X)/q(X)\in E$, with $p, q$ coprime, $q(X)\neq0$, and $\deg(q)>0$, then there exists a polynomial of degree $m>0$ in $E$?
This way, we would demonstrate that $E$ indeed contains no non-constant rational functions and, therefore, $E=F$.
