# Procedure to construct number rings of number fields

I need a good reference to learn the procedure of constructing number rings of number fields. I am currently reading Daniel Marcus, the procedure given there is not getting clear and I am finding it a little confusing. Can anyone refer me a material, where this proofs are presented clearly?

• From the primitive element theorem find $$c\in K$$ such that $$K=Q(c)$$. Let $$m$$ be the product of denominators of its minimal polynomial and $$a=mc$$. Then $$a\in O_K$$ such that $$K=Q(a)$$.
• Show that $$n O_K\subset Z[a]$$ where $$n=Disc(Z[a])= |O_K/Z[a]|^2 Disc(O_K)=\det(Tr(a^i a^j))$$
• $$n^{-1} Z[a]/Z[a]$$ is a finite group, pick a representative $$b_j$$ of all its element to find which ones are algebraic integers (checking if $$\det(xI-B_j)\in Z[x]$$ where $$B_j$$ is the matrix of the multiplication by $$b_j$$ in the $$[K:Q]$$ dimensional $$Q$$-vector space $$Q[a]$$), you'll have $$O_K= \bigcup_{b_j\in O_K} (b_j+Z[a])$$
• Not too hard to find a $$Z$$-module basis of $$O_K$$ from there.
• Not quite sure about the 2nd and 3rd point's proofs.. My point was I know the proof for quadratic number field clearly, but I did not understand the proof about other number rings, for instance cubic number ring $\mathbb{Z}[a^{1/3}]$ or cyclotomic number rings $\mathbb{Z}[\zeta_r]$, was asking about reading materials for that.. Commented Oct 23, 2020 at 17:50