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Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $ 6 $$6$

Let us consider $ \eta $ beconsider a primitive $ 7^{th} $ roots$7^{\text{th}}$ root of unity then$\eta$. Then the minimal polynomial of $ \eta $ over $ \mathbb{Q} $ is $1 + \eta +.....+ \eta^{6} $$1 + \eta +.....+ \eta^{6}$.So So the dimension of the $\mathbb{Q} $ space$\mathbb{Q}$-space $ \mathbb{Q}(\eta) = L$  (say) is $ 6 $ and .Now Now the group $G=Gal(\mathbb{Q}(\eta)/\mathbb{Q}) $$G=\operatorname{Gal}(\mathbb{Q}(\eta)/\mathbb{Q}) $ is a cyclic group of order $ 6 $ and let $ \sigma $ be thea generator.Let Let $ a \in L^{\times} $ and $ b,c \in L_{3} $  ,where where $ L_3 $ is the fixed field of $ \sigma^{3} $  . Consider the equation $ \sigma^{2}(a)/(a) = b/c - 1 $ . Now what$$ \frac{\sigma^{2}(a)}{a} = \frac{b}{c} - 1 .$$ 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 greatargreater than $3$ since if $ b=0 $ or $ c=0 $ or $ b=c $ then it has no solution  ,in in all cases the dimension of the corresponding subspaces is $ 3 $. Also the eigenvalues of $ \sigma^{2} $ are $ 1,\omega ,\omega^2 $.

Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $ 6 $

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 $.

Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $6$

Let us consider a primitive $7^{\text{th}}$ root of unity $\eta$. 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 $. Now the group $G=\operatorname{Gal}(\mathbb{Q}(\eta)/\mathbb{Q}) $ is a cyclic group of order $ 6 $ and let $ \sigma $ be a 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 $$ \frac{\sigma^{2}(a)}{a} = \frac{b}{c} - 1 .$$ 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 greater 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 $.

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Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $ 6 $

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 $.