When computing the ideal class group of a quadratic extension $\mathbb{Q}[\alpha]$ after we have decomposed all rational primes smaller than the Minkowski bound into generating prime ideals $\mathfrak{p}_1,\dots\mathfrak{p}_n$ it is common to look at the (rational) prime decomposition of some $N(k+\alpha)$ (for small $k$) and then deduce that $(k+\alpha)$ decomposes into the corresponding $\mathfrak{p}$ (e.g. if 2 divides $N(k+\alpha)$ and $(2)=\mathfrak{p}_2\bar{\mathfrak{p}_2}$ then either $\mathfrak{p}_2$ or its inverse appear in the decomposition of $(k+\alpha)$).
Question: what justifies this step?