Let $K$ be a number field with ring of integers $\mathcal{O}$ containing a non-principal ideal $\mathfrak{i}$.
Let us find generators $(a_1 ... a_n)$ for $\mathfrak{i}$ such that $N(\prod_{i=1}^na_i)$ is minimal. We have found a "small" basis with respect to the norm. Note that this basis is not necessarily unique (even up to units).
Now let us define the ratio $N(a_1): ... :N(a_n)$ for this ideal. I want to find out when this ratio is well defined for every ideal in the number ring. In particular it would be useful to see if this is true in quadratic number rings, where it seems to be true.
Example: consider the ideal $(7, 3 + \sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$. This basis can be reduced to $(4-\sqrt{-5}, 3+\sqrt{-5})$ where $N((4-\sqrt{-5})(3+\sqrt{-5})) = N(21 \cdot 14) = 294$ is minimal. In this case the ratio is $21 : 14 = 3: 2$. Ordering in these ratios is not important.
Context: This ratio (or ratios in the case of failure) are preserved under multiplication by principal ideals. Hence every ideal class in the class group has a ratio/s assigned to it. Automorphisms preserve ratios so two classes can have the same ratio, but it could still be an interesting approach to the ideal class group.
Experience: Go easy on the local fields, i'm still a beginner there. Everything else should be ok.