Let $E$ be a field. If $G$ is subgroup of $\text{Aut} (E)$, then the set: $\text{Inv}(G) = \{a \in E: \forall \ \eta \in G, \eta(a) = a\}$ is a subfield of $E$. Artin's lemma is:
Let $E$ be a field, $G$ be a finite subgroup of $\text{Aut}(E)$, and $F = \text{Inv}(G)$. Then, $[E:F] \le |G|$.
The proof in the textbook is not very satisfying. It is not clear what is going on. I completely understand it, but it seems to come out of nowhere.
Can someone provide some intuition?
Here's the proof: ($(17)$ is the equation $[E:F] \le |G|$)
(Just ignore the bit about linear equations..)