Discrete logarithms in quasi-polynomial time in finite fields of fixed characteristic

We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. More generally, we prove that it can be solved in the field of cardinality p n p^n in expected time ( p n ) 2 log 2 ⁡ ( n ) + O ( 1 ) (pn)^{2\log _2(n) + O(1)} .