| 11. | All finite groups do occur as Galois groups.
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| 12. | The automorphisms also leave fixed, so they are elements of the Galois group.
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| 13. | Therefore, if belongs to the Galois group of, then maps into itself.
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| 14. | Galois groups for infinite extensions are profinite groups.
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| 15. | This means that the number of elements in the associated Galois group is 1.
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| 16. | This interpretation of the Galois group is known as Grothendieck's Galois theory.
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| 17. | This group is isomorphic to the absolute Galois group of an arbitrary finite field.
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| 18. | The Galois group of this septic is the maximal solvable group of order 42.
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| 19. | See also Complex conjugate and Galois group.
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| 20. | There is even a polynomial with integral coefficients whose Galois group is the Monster group.
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