I have a question about a step in the proof of following statement:
Let $A$ be an integrally closed domain with the field of fractions $K, L$ a finite normal extension of $K, B$ the integral closure of $A$ in $L$. Then the group $G=\operatorname {Gal}(L/K)$ acts transitively on each fiber of $\operatorname {Spec}B\to \operatorname {Spec}A$.
Here one considers the prime ideals $p_1$ and $p_2$, and $\sigma \in \operatorname {Gal}(L/K)$
Suppose $p_2\neq \sigma(p_1)$. Why then the prime avoidance provides that there is an element $x \in p_2$ such that $\sigma (x)\not \in {\mathfrak {p}}_{1} $ for any $\sigma$? Thats not clear to me.
The statement of prime avoidance is that if an ideal $I$ in a commutative ring $R$ is contained in a union of finitely many prime ideals $P_i$'s, then it is contained in $P_i$ for some $i$.
Source: https://en.wikipedia.org/wiki/Integral_element#Integral_extensions