Trivial Field Extension is Galois

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Theorem

Let $F$ be a field.


The trivial field extension $F / F$ is Galois.


Proof

We shall show Definition 1 of Galois Extension.

Observe:

\(\ds \Gal {F / F}\) \(=\) \(\ds \set {\sigma \in \Aut F: \forall k \in F: \map \sigma k = k}\) Definition of Galois Group of Field Extension
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \set {I_F}\)

where $I_F$ denotes the identity mapping on $F$.

Therefore:

\(\ds \map {\operatorname{Fix}_F} {\Gal {F / F} }\) \(=\) \(\ds \set {f \in F : \forall \sigma \in \Gal {F / F} : \map \sigma f = f}\) Definition of Fixed Field
\(\ds \) \(=\) \(\ds \set {f \in F : \map {I_F} f = f}\) by $(1)$
\(\ds \) \(=\) \(\ds F\)

$\blacksquare$