What is the computational complexity (in terms of big $O()$ notation) of arithmetic in the finite field $\mathbb{F}_{2^k}$ or alternatively is there a good reference?
I am interested in the asymptotic computational complexity of addition, multiplication and inverse.
With the caveat that you're probably asking the wrong question, general purpose methods for doing arithmetic with polynomials over finite fields tends to run directly parallel to integer methods; e.g. multiplication of large polynomials via Schönhage-Strassen and division by Barrett's algorithm and using newton's algorithm for inverses.
And general purpose computation in large finite fields is simply modular arithmetic.