I am reading Algebraic Number Fields by Gerald Janusz. Let $K$ be a number field with number ring $R$. In chapter IV prop 4.3, he tries to apply zeta functions to prove there are infinitely many prime ideals in a number field with relative degree 1 over $Q$. Let this set be $S$.
In the proof, he begins with "if we exclude the finite number of ramified primes which may be in $S$, then $S$ is the set of primes $B$ for which $N(B)=p$ is prime".
My question is, the relative degree $f$ of prime $B$ over $Q$ is the degree of field extension of $\mathbb{Z}/(B\cap \mathbb{Z}) \subset R/B$. Let $|\mathbb{Z}/(B\cap \mathbb{Z})|=p$ and $|R/B|=p^f=N(B)$. Then $f=1$ should be equivelent to that $N(B)$ is an integer prime. Why does Janusz try to first exclude ramified primes in $S$ to conclude $S$ are those primes with prime index?
This is the complete proof in the book if anyone needs. Thank you.
