| 1. | See the article on Galois groups for further explanation and examples.
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| 2. | The field norm can also be obtained without the Galois group.
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| 3. | The Galois group is cyclic of order and is generated by.
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| 4. | Extensions whose Galois group is abelian are called abelian extensions.
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| 5. | Since divides, the Galois group has a cyclic subgroup of order.
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| 6. | The absolute Galois group is well-defined up to inner automorphism.
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| 7. | If is a finite Galois extension of, the Galois group is solvable.
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| 8. | As the Galois group of is abelian, this is an abelian extension.
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| 9. | It follows that has Galois group isomorphic to over.
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| 10. | The Galois group is naturally isomorphic to the multiplicative group
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