Confusion in notation, Action of $Gal(\bar{K}/K)$

Question

On page 2, Line 1 Book says $Gal(\bar{K}/K)$ acts on $A^{n}$, how this action is defined.

What does this notation $P^{\sigma}$ means?

Answer 1

$P^{\sigma}$ is simply $\sigma$ acting on $P$. $P = P^{\sigma}$ means that $P$ is invariant under the action of the Galois group.