Primitive roots as linear combination of a $\mathbb{Q}$-basis of $\mathbb{Q}(\epsilon)$

Question

Let $\epsilon$ be a 9-primitive root of unity

I got that the $\mathrm{Irr}(\epsilon,\mathbb{Q})=x^6+x^3+1$ so a $\mathbb{Q}$-basis of $\mathbb{Q}(\epsilon)$ is $\{1,\epsilon,\epsilon^2,\epsilon^3,\epsilon^4,\epsilon^5\}$.

I know that all the primitive roots are $\epsilon^r$ with $1\le r<n$ and $\gcd(r,n)=1$, so those are $\epsilon,\epsilon^2,\epsilon^4,\epsilon^5,\epsilon^7,\epsilon^8$.

The problem I have now is I don't know how to express for example $\epsilon^7$ as a linear combination of the base elements.

Any hints?

Answer 1

Multiply the equation $$ \epsilon^6 + \epsilon^3 + 1 = 0 $$ by $\epsilon$ and solve for $\epsilon^7$.

Alternatively substitute $\epsilon^6 = - \epsilon^3 - 1$ into $\epsilon^7 = \epsilon(\epsilon^6)$.