21. In certain situations, the Galois group dualities, such as Poitou-Tate duality. 22. If a polynomial is irreducible, then the corresponding Galois group is a transitive subgroup. 23. When the extension is Galois this automorphism group is called the Galois group of the extension. 24. A field extension is called a cyclic extension if its Galois group is a cyclic group. 25. He introduced the concept of geometric Galois representation of the Galois group of a number field. 26. Generalizing this argument shows that the Galois group of every general polynomial of degree is isomorphic to. 27. Therefore, it must be an even number, and so the Galois group can only be. 28. So, in this case, the Galois group of is not and therefore it must be. 29. The construction in the preceding section used these generators to establish a polynomial's Galois group . 30. Consequently, it is a topological generator in the usual Krull topology on the absolute Galois group .
