What kinds of non-zero characteristic fields exist? There are these finite fields of characteristic $p$ , namely $\mathbb{F}_{p^n}$ for any $n>1$ and there is the algebraic closure $\bar{\mathbb{F}_p}$. The only other fields of non-zero characteristic I can think of are transcendental extensions namely $\mathbb{F}_{q}(x_1,x_2,..x_k)$ where $q=p^{n}$. 
Thats all! I cannot think of any other fields of non-zero characteristic. I may be asking too much if I ask for characterization of all non-zero characteristic fields. But I would like to know what other kinds of such fields are possible.
Thanks. 

The basic structure theory of fields tells us that a field extension $L/K$ can be split into the following steps:


*

*an algebraic extension $K^\prime /K$,

*a purely transcendental extension $K^\prime (T)/K^\prime$,

*an algebraic extension $L/K^\prime (T)$.


The field $K^\prime$ is the algebraic closure of $K$ in $L$ 
and thus uniquely determined by $L/K$.
The set $T$ is a transcendence basis of $L/K$; its cardinality
is uniquely determined by $L/K$.
A field $L$ has characteristic $p\neq 0$ iff it contains the
finite field $\mathbb{F}_p$. Hence you get all fields of characteristic
$p$ by letting $K=\mathbb{F}_p$ in the description of field extensions,
and by chosing $T$ and $K^\prime$ and $L/K^\prime (T)$ as you like.
Of course in general it is then hard to judge whether two such fields
are isomorphic - essentially because of step 3. 
