Is there a general method of constructing fields of the form $\operatorname{GF}(2^k)$? (preferably something that is easily manipulable by a computer.)
I know that one can look for an irreducible polynomial of degree $k$ over $\mathbb{Z}_2$, but AFAIK finding one is not a simple task (for very large $k$).
Thank you.
Finding an irreducible polynomial (try random polynomials, and check whether they are irreducible) actually is not hard. (See How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?
for a count.) If your $k$ is so large that testing polynomials for efficiency becomes an issue, most likely field arithmetic itself would be an issue as well.
In practice the main issue (not least for efficient arithmetic) is to ensure that field extensions are embedded compatibly (E.g. when working in $GF(2^3)$ and $GF(2^4)$ have the sum of elements lie in $GF(2^{12})$), and to ensure that different persons working over the field can construct the same element.
The technique for achieving this compatibilities is to use so-called Conway Polynomials: http://en.wikipedia.org/wiki/Conway_polynomial_(finite_fields)
