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I know, if $G$ ist the cyclic group of order $p$ ($p$ odd), that the simple $\mathbb{Q}G$ modules are $\mathbb{Q}$ and $\mathbb{Q}(\zeta_p)$. In that way I get the irreducible representations of the cyclic group $C_p$ over $\mathbb{Q}$.

Is there a similar way to get the irreducible representations of the dihedral group $D_p$ with $2p$ Elements over $\mathbb{Q}$, or rather all simple $\mathbb{Q}D_p$ modules?

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Regard $C_{p}$ as a normal subgroup of index $2$ of $D_{p}.$ Then the trivial $\mathbb{Q}C_{p}$-module extends in two ways: in one of these, the involution acts trivially, and in the other it is represented by $-1.$ The trick for the other irreducible representation of $\mathbb{Q}C_{p}$ is to notice that it extends uniquely to a representation of $\mathbb{Q}D_{p}.$ – Geoff Robinson Apr 9 '12 at 11:50

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