# Isomorphism map from $\Bbb{Q}(C_2\times C_2)$ to $\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q}$

I know how to find structure of Rational group algebras of finite cyclic groups such as for cyclic group $C_6=\langle a \rangle$ we have $\Bbb{Q}C_6 \cong \Bbb{Q} \oplus \Bbb{Q}\ \oplus\ \Bbb{Q}(\omega)\ \oplus \Bbb{Q}(\omega^2)$ where $a \to (1,1,\omega, \omega^2)$.

But when we are finding structure over finite abelian groups which are not cyclic, like $C_2 \times C_4$, then by perlis walker theorem, I can find that $\Bbb{Q}(C_2\times C_2) \cong \Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q}$ but how do I determine what is the isomorphism and where do I send which elements. The proof is by induction and does not reveal anything specific in case of non cyclic groups.

Thanks.

Similarly how to find what does $\Bbb{Q}(C_2\times C_4)$ is mapped into as isomorphism.

• Sehgal book may helpful....introduction to group rings..... – neelkanth Jan 7 '16 at 15:29
• @neela I am doing that book – Bhaskar Vashishth Jan 7 '16 at 16:42
• Doing research on group rings? – neelkanth Jan 8 '16 at 2:06
• Yes. Doing my MS thesis – Bhaskar Vashishth Jan 10 '16 at 6:46