31. Every projective profinite group can be realized as an absolute Galois group of a pseudo algebraically closed field. 32. There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups . 33. This induces canonical continuous actions of the absolute Galois group of " K " on the lattices. 34. However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of order eight. 35. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins. 36. The Galois group of a polynomial of degree n is S _ n or a proper subgroup of that. 37. A Picard Vessiot extension is Liouvillian if and only if the connected component of its differential Galois group is solvable. 38. In terms of Galois theory, this means that is a Galois extension of, which has a cyclic Galois group . 39. This Galois group has only two elements : \ sigma \, and the identity on \ mathbb { C }. 40. Galois cohomology makes no assumption that Galois groups are abelian groups, so that this was a non-abelian theory.
