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.