Automorphism of a field extension

Let $𝐿 : 𝐾$ be a field extension. An automorphism $𝜑 \in A u t (𝐿 : 𝐾)$ of $𝐿 : 𝐾$ is a field automorphism of $𝐿$ fixing (the image of) $𝐾$ pointwise, #m/def/field i.e.

A u t (𝐿 : 𝐾) = {𝜑 \in A u t (𝐿) : 𝜑 ↾ 𝐾 = i d 𝐾}

In the case of a Galois extension, this is denoted $G a l (𝐿 : 𝐾)$ 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

#state/tidy | #lang/en | #SemBr