Galois cohomology of fields with a dimension

We consider fields K with an abstract notion of dimension as stated by Pillay and Poizat in their paper of 1995. We prove that, for every finite extension L of K and for every finite Galois extension L1 of L, the Brauer group Br(L1/L) is finite. Moreover, given an algebraic group G defined over L, we have that H1(L1/L,G) is finite.