TITLE: Wedderburn's theorem for $\mathbb{Q}G$
QUESTION [12 upvotes]: Let $G$ be a finite group and let $\mathbb{Q}G=M_{n_1}(D_1)\times\cdots\times M_{n_k}(D_k)$ be the decomposition of $\mathbb{Q}G$ as a product of rings of matrices over divisions rings. Let $Z_i$ be the center of $D_i$. Then each $Z_i$ is an algebraic extension of $\mathbb{Q}$. The question is: how much is it known about these algebraic number fields?. More precisely, for which family of groups $G$ it is known that each $Z_i$ is a Galois extension of $\mathbb{Q}$ and in such a case what can we say about the corresponding Galois groups?.
Any comments and references will be strongly appreciated.

REPLY [2 votes]: The simplest case of your question is the case where $G$ is the cyclic group of order $n$, it is known
that $\mathbb{Q}G\simeq\mathbb{Q}[X]/(X^n-1)$.
As $X^n-1=\Pi_{d|n}\Phi_d(X)$ where $\Phi_d(X)$  is the $d$-th
cyclotomic polynomial, it follows that
$\mathbb{Q}G\simeq\Pi_{d|n}\mathbb{Q}(\zeta_d)$ where $\zeta_d\in\mathbb{C}$ is a
primitive $d$-th root of unity.
Edit: This is also a very particular but concrete case of the idea of
@Aakumadula.