What is an Algebraic function field?

Question

I can't understand what is an Algebraic function field.

1. Some references says that, $F/K$ is an Algebraic function field if it contains at least one transcendental number.

2. I found and example for a field of positive characteristic $p$: $K=\mathbb{F}_q(t_1,...,t_k)[\alpha_1,...,\alpha_r]/I$, where $I$ is a maximal ideal of the polynomial ring $\mathbb{F}_q(t_1,...,t_k)[\alpha_1,...,\alpha_r]$ and $q$ is a power of $p$. Is that an example for a Algebraic function field?

3. Can some one give a simple example for that. I know $\pi$ and $e$ are transcendental. Can I use them to construct an Algebraic function field? For example:$\mathbb{Q}(\pi)$, or $\mathbb{Q}(\pi)[x]/<x^2+5>$ ?.

4. Also, I can' t see how elements of the Algebraic function field looks like. Is that like, if I take $a=\sqrt{-5}$ for the example $\mathbb{Q}(\pi)[x]/<x^2+5>$, one elements maybe, $4\pi+ \frac{2}{3}{\pi}^2+a$?

Please help me to understand this. Thank you.

Answer 1

An algebraic function field $F$ in $n$ indeterminates over a field $K$ is simply a finite extension of the field $K(X_1, \dots, X_n)$ of rational fractions over $K$ (or an extension field isomorphic to such a field). $n$ is its transcendence degree over $K$.