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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. |
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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. |
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