In the representation theory of finite groups in characteristic 0, we have that the group ring completely decomposes into a product of matrix algebras once tensored with $\mathbb{C}$, and similarly over any algebraically closed field of characteristic 0.
I recall reading somewhere that it is a (deep) theorem of Brauer that the group algebra decomposes completely over $\mathbb{Q}(\zeta_{|G|})$, where $\zeta_{|G|}$ is a primitive $|G|$th root of unity.
Does anyone have a reference for this result, and/or an idea of the techniques used in the proof?