Let $F$ be an extension field over $K$ , if $[F(x):K(x)]$ is finite , then is $F$ also a finite extension over $K$ ?

Question

Let $F$ be an extension field over $K$ such that $F(x)$ is a finite extension over $K(x)$ ; then is it true that $[F:K]$ is also finite ? ( I know about the converse , that if $F/K$ is a finite extension then so is $F(x)/K(x)$ and $[F:K]=[F(x):K(x)]$ holds ) Please help . Thanks in advance

Answer 1

In general, if $V$ is a $K$-vector space and $E/K$ is a field extension, then $\dim_E(V\otimes_K E) = \dim_K V$.

Apply that to $V = F$ and $E = K(x)$.

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An elementary version : suppose $[F:K]$ is infinite, and $(x_i)_{i\in I}$ is an infinite family of linearly independent elements of $F$. Suppose they have a linear relation in $F(x)$ over $K(x)$, and try to reduce to the case $\sum_i P_ix_i = 0$ with $P_i\in K[x]$ such that the $P_i$ are not all divisible by $X$.

Then try to conclude.