41. These permutations together form a permutation group, also called the Galois group of the polynomial ( over the rational numbers ). 42. It is known that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. 43. It is known that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. 44. If the Galois group is supersolvable or more generally monomial then all representations are of this form so the Artin conjecture holds. 45. One resolvent can tell if the Galois group of a polynomial is a ( not necessarily proper ) subgroup of given group. 46. Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of "'Q " '. 47. Sheaves endowed with nontrivial endomorphisms, such as the action of an algebraic torus or a Galois group , are of particular interest. 48. ;"'Frobenius field "': A pseudo algebraically closed field whose absolute Galois group has the embedding property. 49. Then the question is this : is there a Galois extension field such that the Galois group of the extension is isomorphic to? 50. After that, one applies Hilbert's irreducibility theorem to specialise, in such a way as to preserve the Galois group .
