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