State the Artin–Wedderburn theorem for finite-dimensional semisimple algebras, and describe the rational group algebras of \(C_5\), \(C_2\times C_2\), and \(Q_8\).
Solution
The Artin–Wedderburn theorem says that every finite-dimensional semisimple algebra over a field is isomorphic to a finite product \[ \prod_i M_{n_i}(D_i), \] where each \(D_i\) is a finite-dimensional division algebra over the base field.
Since \(\QQ\) has characteristic \(0\), Maschke’s theorem makes \(\QQ[G]\) semisimple for every finite group \(G\).
1. \(G=C_5\).
Using \[ \QQ[C_5]\cong \QQ[x]/(x^5-1) \] and \[ x^5-1=(x-1)\Phi_5(x), \] with \(\Phi_5\) irreducible over \(\QQ\), the Chinese remainder theorem gives \[ \QQ[C_5]\cong \QQ\times\QQ(\zeta_5). \]
2. \(G=C_2\times C_2\).
All four irreducible characters are one-dimensional and rational-valued. Hence \[ \QQ[C_2\times C_2]\cong \QQ^4. \]
3. \(G=Q_8\).
The quaternion group has four one-dimensional rational representations and one quaternionic irreducible component. Accordingly, \[ \QQ[Q_8] \cong \QQ^4\times(-1,-1)_{\QQ}, \] where \((-1,-1)_{\QQ}\) is the rational Hamilton quaternion division algebra \[ \QQ\langle i,j\mid i^2=j^2=-1,\ ij=-ji\rangle. \] The dimensions are \(4+4=8\), as required.