For
q
q
a prime power, the discrete logarithm problem (DLP) in
F
q
\mathbb {F}_{q}
consists of finding, for any
g
∈
F
q
×
g \in \mathbb {F}_{q}^{\times }
and
h
∈
⟨
g
⟩
h \in \langle g \rangle
, an integer
x
x
such that
g
x
=
h
g^x = h
. We present an algorithm for computing discrete logarithms with which we prove that for each prime
p
p
there exist infinitely many explicit extension fields
F
p
n
\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
F
p
n
\mathbb {F}_{p^n}
in expected quasi-polynomial time.