Problem when computing ideal class group

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?

• Uniqueness of factorization of ideals into a product of prime ideals combined with the fact that the norm of a prime ideal is a power of a rational prime. More precisely, of the rational prime it contains. – Jyrki Lahtonen May 3 '16 at 9:15