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**tangibleLime** Problem:

There is an isomorphism of $\displaystyle U_{7}$ with $\displaystyle \mathbb{Z}_{7}$ in which $\displaystyle \zeta = e^{i(2pi/7)} \leftrightarrow 4$. Find the element in $\displaystyle \mathbb{Z}_{7}$ to which $\displaystyle \zeta^{m}$ must correspond for $\displaystyle m = 0, 2, 3, 4, 5, 6$.

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After arriving at an answer, I noticed that I said there was no answer for all of the even m values, so I'm not sure if I am correct.

To figure out the isomorphism, I used the example in the problem,

$\displaystyle \zeta^{1} \leftrightarrow 4$.

I reasoned that $\displaystyle 4 + 4 \in \mathbb{Z}_{7} = 1$, in which that 1 corresponds to the exponent of zeta.

I used this to look for the other values asked in the problem.

For m=0, $\displaystyle 3.5 + 3.5 \in \mathbb{Z}_{7} = 0$, but $\displaystyle 3.5 \notin Z$, so I said that for m=0, there is no corresponding zeta function.

For m=3 (skipping ahead), $\displaystyle 5 + 5 \in \mathbb{Z}_{7} = 3$, $\displaystyle 3 \in \mathbb{Z}_{7}$, so $\displaystyle \zeta^{3} \leftrightarrow 5$.

I think I'm doing something incorrectly. Because since this is an isomorphism, and therefore a one-to-one correspondence, shouldn't all m=0,1,2,3,4,5,6 in $\displaystyle U_{7}$ map to a member of $\displaystyle \mathbb{Z}_{7}$? My method is producing answers for only the odd numbers.

Any help is extremely appreciated.