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 $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds F\) |
$\blacksquare$