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. 



Wedderburn's theorem for $\mathbb{Q}G$
The short answer is yes, the centres $Z\_i$ are contained in the cyclotomic extension $E\_n= {\mathbb Q}(e^{(2\pi i)/n})$ where $n$ is the order of the group. As pranavk says, we need only prove that the simple factors of the centre of ${\mathbb Q}[G]$ are contained on $E\_n$. But the centre is spanned by the averages $ C\_x =\sum gxg^{-1}$ where the sum is over all the elements of $G$. Given an absolutely irreducible representation $V$ of $G$ of dimension $r$ , each $C\_x$ acts by a scalar $\lambda $ say. By taking traces, you see that $\lambda $ is a rational multiple of the trace of $x$; since $x$ has order dividing $n$, all its eigenvalues in $V$ are $n$-th roots of unity, and hence the trace of $x$ lies in $E\_n$. 


Now,in your notation, each $M\_{n\_i}(D\_i)$ after tensoring (over $Z\_i$) with the algebraic closure of ${\mathbb Q}$ is of the form $End (V)$ for some absolutely irreducible representation $V$ of $G$, hence each $Z\_i$ (which as pranavk observed, lies in the centre of ${\mathbb Q}[G]$) lies in $E\_n$. 


I believe all this is worked out in Serre's book on finite groups (Springer notes) and is well known to experts (I am not one!). 


