Let $L/K$ be finite galois number field extension and $p$ be prime of (ring of integers of) $K$. Let $G=Gal(L/K)$.
It is well known that $G$ acts transitively on the set of all prime ideals of $L$ above $p$.
The proof of this fact I know is proof by contradiction. It goes on like, If we assume there exists $B_1,B_2$, for all $ \sigma \in G$, $ B_1≠\sigma(B_2)$, then from Chinese reminder theorem there exists $x \in O_K$ s.t $x≡0modB_2$ and $x≡1mod \sigma B_1$. $N_{L/K}(x)$ is in $p$, on the other hand, from hypothesis of proof by contradiction, $x$ cannot be in $p$, contradiction▪️
My question is, can't we prove by finding explicit $ \sigma \in G$, which sends one ideal over $p$ to another prime above $p$ ?