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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.

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    $\begingroup$ Hint: given two elements of $\mathbb{F}_7^*$, how could you tell if they represent the same element in the quotient group? Remember, this is the same as asking if they lie in the same left coset of $\mathbb{F}_7^{*3}$. $\endgroup$ – diracdeltafunk Jun 2 at 12:42
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    $\begingroup$ Thanks! I just wrote everything down and it all followed nicely. $\endgroup$ – Math4Life Jun 2 at 13:09
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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 $1,3,9$.

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