Let $x$ be transcendental over $F$. Let $y=f(x)/g(x)$ be a rational function. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$

Solution 1:

Hint : Since $x$ is transcendental over $F$, we get that $y$ is also transcendental. In particular, $F[y]$ is a UFD, therefore to show that $R$ is irreducible in $F(y)[t]$, it is enough to show that it is irreducible in $F[y][t]$ by Gauss' lemma.

But then $F[y][t] = F[t][y]$, so you can act like $y$ is a variable and we can take the $y$ degrees during comparison of equality in this integral domain. What is the $y$ degree of $R(t)$?

To finish, what is the $t$ degree of $R$?

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