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. 

No need to limit yourself to a finite number of transcendentals... So $\mathbb F_q(x_1,x_2,\dots,x_n,\dots)$ is another example. You can also use $\bar{\mathbb{F}_p}$ as the  coefficient field. Many combinations are possible. What characterization are you after?
