Field theory MOC

Galois extension

An extension is Galois iff it is separable and normal. gal

Finite Galois extension

Let be a finite extension. Then the following are equivalent:¹

1. is Galois;
2. is the splitting field of a separable polynomial over ;
3. is separable and normal;
4. ;
5. is the fixed field for ;
6. the Galois correspondence for is a bijection;
7. is separable, and if is an algebraic extension and , then .

Properties

• Every extension of a Galois field is Galois cyclic (generated by the Frobenius automorphism)

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develop | en | sembr

Footnotes

1. 2009. Algebra: Chapter 0, §VII.6.1, p. 457 ↩