Definition of finite field with fixed characteristic In this article Discrete logarithms in quasi-polynomial time in finite fields of fixed characteristic the term finite fields of fixed characteristic is not defined and I couldn't find it on the literature, too.

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*What is the definition of finite fields of fixed characteristic

The term quasi-polynomial time means quasi-polynomial in...which parameters? What fixed characteristic says is that the problem is quasi-polynomial in the field size as the field size varies, as long as we keep the characteristic fixed.
So, for instance, if their algorithm takes $\le C2^pk^r$ steps over a field of characteristic $p$ and size $k$ for some constants $C,r$, then it is polynomial for fields of fixed characteristic, but exponential in the characteristic.
