## Computation in cyclotomic field $\mathbb{Q}(\zeta_{7})$ over $\mathbb{Q}$.

I have some question about computation in cyclotomic field $$K=\mathbb{Q}(\zeta)$$, where $$\zeta$$ is a primitive $$7$$th root of unity.

I know that the subfield $$E=\mathbb{Q}(\zeta+\zeta^{2}+\zeta^{4})$$ of $$\mathbb{Q}(\zeta)$$ is of degree $$2$$ over $$\mathbb{Q}$$.

Actually, I found the primitive element of $$E$$ as $$\zeta+\zeta^{2}+\zeta^{4}$$ using the fact that $$E$$ is the fixed field of $$\langle\sigma\rangle$$, where $$\sigma(\zeta)=\zeta^{3}$$.

Now, considering $$E$$ as a vector space over $$\mathbb{Q}$$, then $$E$$ has a $$\mathbb{Q}$$-basis as $$\{1,\zeta+\zeta^{2}+\zeta^{4}\}$$. (Is it possible?)

If it possible, how to write certain elements, for example, $$\zeta^{3}$$ and $$\zeta^{6}$$, as a linear combination of elements of the previous $$\mathbb{Q}$$-basis?

I tried to use the fact that $$\zeta^{7}=1$$ and $$\Phi_{7}(\zeta)=0$$, but i can’t find any relation.

Can anyone help me? Thank you.