On the discrete logarithm problem in finite fields of fixed characteristic

For~$q$ a prime power, the discrete logarithm problem (DLP) in~$\F_{q}$ consists in finding, for any $g \in \mathbb{F}_{q}^{\times}$ and $h \in \langle g \rangle$, an integer~$x$ such that $g^x = h$. We present an algorithm for computing discrete logarithms with which we prove that for each prime~$p$ there exist infinitely many explicit extension fields~$\mathbb{F}_{p^n}$ in which the DLP can be solved in expected quasi-polynomial time. Furthermore, subject to a conjecture on the existence of irreducible polynomials of a certain form, the algorithm solves the DLP in all extensions~$\mathbb{F}_{p^n}$ in expected quasi-polynomial time.