I'm studying Algebraic Function Fields and Codes book from Henning Stichtenoth and I didn't understand this remark in the first page:
I couldn't solve any part of the equivalence, I think maybe because I don't remember very well the Galois theory classes, if someone could help in this remark I would be very grateful.
I need help.
Thanks in advance.
Thanks to Jyrki's comment we can proceed. If $F/K(z)$ is algebraic, then $K(z)$ is of infinite degree, hence transcendental by definition since $[F:K]\le [F:K(z)][K(z):K]$ and we know $[F:K]=\infty$ and $[F:K(z)]<\infty$, by the finite generation and algebraic hypotheses it must be that $[K(z):K]=\infty$.
On the other hand, if $z\in F$ is transcendental, then $[K(z):K]=\infty$ and by definition of transcendence degree, the longest chain of pairwise infinite degree extensions we can have with initial field $K$ and final field $F$ is of length $1$, hence $K\subseteq K(z)\subseteq K(z, \alpha_1)\subseteq \ldots\subseteq K(z,\alpha_1,\alpha_2,\ldots,\alpha_k)$ must have $[K(z,\alpha_1,\ldots,\alpha_k):K(z)]<\infty$ for any $k$ and $\alpha_i\in F$ since the first inclusion has infinite degree. But since $F$ is finitely generated, $F=K(z,\alpha_1,\ldots, \alpha_n)$ so that the $\alpha_i$ must be algebraic over $K(z)$ since that first extension is infinite degree. Hence $F/K(z)=K(z,\alpha_1,\ldots, \alpha_n)/K(z)$ is algebraic.