Specific example of a certain claim in a paper In a paper of Heath-Brown, it is stated on page 24 that for any number field $K$, there exist ideals $\mathfrak{a_1, \dots, a_t}$ s.t. all fractional ideals can be written as the product of powers of the $\mathfrak{a_j}$ and a principal ideal uniquely (with some restrictions on the exponents). What would an example of $\mathfrak{a_1, \dots, a_t}$ be in the case $K = \mathbb{Q}(\sqrt{-5})$?
 A: The class number is 2, so take $t = 1$ and any non-principal ideal would do.
A: I just want to point out that, as Hunter hinted and Heath-Brown mentioned in the paper, the claim follows from the fact that the ideal class group is finite.
Let me outline a proof that would allow you to answer your own question. I'll be consistent with the notation from the claim (quoted below).

The class group of $K$ being finite, one can choose ideals $\mathfrak a_1,\dots,\mathfrak a_t$ so that every (non-zero) fractional ideal $\mathfrak A$ has a unique decomposition $$\mathfrak A=(\alpha)\mathfrak a_1^{l_1}\dots\mathfrak a_t^{l_t},\quad \alpha \in K^\times,\quad l_j \in \mathbf Z,\quad 0\leq l_j < h_j;$$ let $$\mathfrak a_j^{h_j} = (\alpha_j),\quad  1 \leq j \leq t, \quad h(K) = \prod_{j=1}^th_j.$$



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*By the structure theorem for finitely generated abelian groups, the ideal class group is isomorphic to $$\frac{\mathbf Z}{h_i\mathbf Z} \oplus \dots \oplus \frac{\mathbf Z}{h_t\mathbf Z},$$ for suitable positive integers $h_1,\dots h_t$. So how could we choose $\mathfrak a_i$? 



 Let $\mathfrak a_i$ be the ideal whose class in the ideal class group corresponds to $\mathbf Z/ h_i\mathbf Z$. 



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*The class of $\mathfrak a_i^{h_i}$ is trivial in the ideal class group. So how could we choose $\alpha_i$?



 Being trivial in the ideal class group means that $$\mathfrak a_i^{h_i} = (\alpha_i),$$ for some $\alpha_i \in K^\times$.



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*How can we write the class number, $h(K)$, in terms of $h_i$, $i = 1,\dots , t$?



 $h(K) = h_1 \dots h_t$.



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*Finally, how would you choose $\alpha$ and the $l_i$ in the factorization of $\mathfrak A$?



 Choose $l_j$ such that the class of $\mathfrak A$ in the ideal class group is $$\prod_{j = 1}^t[\mathfrak a_j]^{l_j}.$$

