Field theory MOC

Galois extension

An extension 𝐹 :𝐾 is Galois iff it is separable and normal. gal

Finite Galois extension

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

1. 𝐹 :𝐾 is Galois;
2. 𝐹 is the splitting field of a separable polynomial 𝑓(𝑥) ∈𝐾[𝑥] over 𝐾;
3. 𝐹 :𝐾 is separable and normal;
4. |Aut⁡(𝐹:𝐾)| =[𝐹 :𝐾];
5. 𝐾 =𝐹Aut⁡(𝐹:𝐾) is the fixed field for Aut⁡(𝐹 :𝐾);
6. the Galois correspondence for 𝐹 :𝐾 is a bijection;
7. 𝐹 :𝐾 is separable, and if 𝐿 :𝐹 is an algebraic extension and 𝜎 ∈Aut⁡(𝐿 :𝐾), then 𝜎(𝐹) =𝐹.

Properties

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

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Footnotes

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