I am stuck at Question and I hope someone can help me.
So, let $G$ be a finite cyclic group of order $n$. I am able to proof the following isomorphisms for the group algebra $\mathbb{Q}G$: $$\mathbb{Q}G \cong \mathbb{Q}[X]/(X^n-1) \cong \mathbb{Q}[X]/\prod_{d|m} \Phi_d(X) \cong \prod_{d|m} \mathbb{Q}[X]/ \Phi_d(X) \cong \prod_{d|m} \mathbb{Q}[\zeta_d]$$ ($\Phi_d$ denotes the $d$-th cyclotomic polynomial).
I found a Therem in "Maximal Orders" by Irving Reiner, which then states, that the maximal orders of $\mathbb{Q}G$ is $\Gamma:=\prod_{d|m} \mathbb{Z}[\zeta_d]$.
Unfortunatly I can not figure out how an embedding of the Integral group ring $\mathbb{Z}G \subseteq \Gamma$ can be defined. My proof for the group algebra doesn't work for $\mathbb{Z}G$. Furthermore I don't believe that $\mathbb{Z}G$ is the maximal order. So there has to be an embedding and some sort of index, but I can't figure out how this might work. Does anyone have an idea or know literature where I might find something about it?
Best reguards
Laura