Let us consider $ \eta $ be a primitive $ 7^{th} $ roots of unity then the minimal polynomial of $ \eta $ over $ \mathbb{Q} $ is $1 + \eta +.....+ \eta^{6} $.So the dimension of the $\mathbb{Q} $ space $ \mathbb{Q}(\eta) = L$(say) is $ 6 $ and .Now the group $G=Gal(\mathbb{Q}(\eta)/\mathbb{Q}) $ is a cyclic group of order $ 6 $ and let $ \sigma $ be the generator.Let $ a \in L^{\times} $ and $ b,c \in L_{3} $ ,where $ L_3 $ is the fixed field of $ \sigma^{3} $ . Consider the equation $ \sigma^{2}(a)/(a) = b/c - 1 $ . Now what is the maximum dimension of a subspace in $ L_3 \oplus L_3 $ such that this equation has no solution.?

Note that the maximum dimension is greatar than $3$ since if $ b=0 $ or $ c=0 $ or $ b=c $ then it has no solution ,in all cases the dimension of the corresponding subspaces is $ 3 $. Also the eigenvalues of $ \sigma^{2} $ are $ 1,\omega ,\omega^2 $.