Galois group of a polynomial

The Galois group of a separable polynomial acts faithfully on its roots; this action is transitive exactly when the polynomial is irreducible.

Suppose a group of field automorphisms acts transitively on elements $y_i$, and each $y_i$ has a unique $p$th root $x_i$ in a stable extension. If every automorphism extends, it sends $x_i$ to the unique root above its image of $y_i$, so the extended group acts transitively on the $x_i$.