Field theory MOC

Automorphism of a field extension

Let 𝐿 :𝐾 be a field extension. An automorphism 𝜑 ∈Aut⁡(𝐿 :𝐾) of 𝐿 :𝐾 is a field automorphism of 𝐿 fixing (the image of) 𝐾 pointwise, field i.e.

Aut⁡(𝐿:𝐾)={𝜑∈Aut⁡(𝐿):𝜑↾𝐾=id𝐾}

In the case of a Galois extension, this is denoted Gal⁡(𝐿 :𝐾) and called the Galois group. An automorphism of a field extension is a special case of an Morphism of field extensions.

Properties

• Bound on the automorphism group of a finite simple extension

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