In Washington's Introduction to Cyclotomic Fields, we are given the following lemma (1.4):
The ideal $(1 - \zeta)$ is a prime ideal of $\mathscr{O}$ and $(1 - \zeta)^{p-1}$= (p). Therefore p is totally ramified in $\mathbb{Q}(\zeta).$
The proof is straightforward for the most part, but I'm a little confused on his alternative proof of the primality.
Let's assume $(1-\zeta) = A \cdot B.$ Then $(p) = N(1-\zeta) = NA \cdot NB.$ This makes sense, and I'm sure I'm missing a simple fact about prime ideals, but why then can we say either NA or NB is 1?