This talk describes several variations of the number field sieve to compute discrete logarithms in finite fields of the form GF(pⁿ), with p a medium to large prime. We show that when n is not too large, this yields a L_pⁿ(1/3) algorithm with efficiency similar to that of the regular number field sieve over prime fields. This approach complements the recent results of Joux and Lercier on the function field sieve. Combining both results, we deduce that computing discrete logarithms have heuristic complexity L_pⁿ(1/3) in all finite fields. To illustrate the efficiency of the algorithm, we provide details of a discrete logarithm computation in a 120-digit finite field F_p³.