Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$ Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$?
That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$?

To a beginner, knowing how one could think of an answer is at least as important as knowing an answer.
For examples in Algebra, one needs (at least) two things: A catalogue of the basic structures that appear commonly in important mathematics, and methods of constructing new structures from old. Your catalogue and constructions will naturally expand as you study more, so you don't need to worry about this consciously. The moral I am trying to impart is the following: Instead of trying to construct a particular a structure with particular properties "from scratch" (like I constantly tried to when I was starting to learn these things), first search your basic catalogue. If that doesn't turn up anything, more often than not your search will hint at some basic construction from one of these examples that will work.
When you start learning field theory, your basic catalogue should include all of the finite fields, the rationals, real numbers, complex numbers, and algebraic numbers. Your basic constructions should be subfields, field extensions, fields of fractions and algebraic closures. You should also have the tools from your basic ring theory; constructions like a quotient ring and ring extensions also help with this stuff. 
For example, Chris came up with his answer by starting with the easy example of a field of characteristic two and wanting to make it bigger. So he extended it with an indeterminate $X$ and as a result he got the field of rational functions with coefficients in $\mathbb{Z}/(2).$ Asaf suggested two ways: making it bigger by taking the algebraic closure, or extending the field by a root of a polynomial (I personally like to see this construction as a certain quotient ring). 
