I'm trying to prove that if $K$ is a countable field and $L:K$ is finitely generated then L is countable.
Based on this post, I know that it follows from the fact that if $K$ is a countable field, then $K(t_1, t_2, ..., t_n)$ is a countable field, but I am unsure of why this is true and how I would show this.
Thank you.