Elements of $\mathbb{F}_7^*/\mathbb{F}_7^{*3}$

Question

I think I have forgotten some basic group theory, but I am having hard time representing the elements from $\mathbb{F}_7^*/\mathbb{F}_7^{*3}$, where $\mathbb{F}_7^{*3}$ denotes all elements that are cubes in $\mathbb{F}_7^*$. I have figured out that $\mathbb{F}_7^{*3} = \{\bar{1},\bar{6}\}$ and hence $\mathbb{F}_7^*/\mathbb{F}_7^{*3}$ is isomorphic to $\mathbb{Z}/3\mathbb{Z}$. However, I am looking for representative elements from $\mathbb{F}_7^*$. Any help would be appreciated.

Answer 1

The group $\Bbb F_7^\ast$, as group of units of a finite field, is a cyclic group and it has order 6. It is generated by every primitive root modulo $7$, for example, 3. Since $3^3\equiv 6\pmod 7$, follows that the quotient group $\Bbb F_7^\ast/\Bbb F_7^{\ast 3}$ has representatives $1,3,9$.