Question

Find all solutions \[\large x^7 \equiv 1 \pmod{7^3}\]

Answer 1

there should be 7 because \[1(\mod7^3) = 1\]

Answer 2

so \[x^7=1\]

Answer 3

yes there are 7 distinct solutions in modulo 7^3

Answer 4

then is the answer not just the 7 solutions of that seventh degree polynomial? Do u want the specific roots?

Answer 5

The roots of \[x^7=1\] are \[x = -(-1)^{\frac{ 1 }{ 7 }}\]\[x = (-1)^{\frac{ 2 }{ 7 }}\]\[x = -(-1)^{\frac{ 3 }{ 7 }}\]\[x = (-1)^{\frac{4 }{ 7 }}\]\[x = -(-1)^{\frac{ 5 }{ 7 }}\]\[x = (-1)^{\frac{ 6 }{ 7 }}\]\[x=1\]

Answer 6

is that what you were looking for?

Answer 7

well, if you want to know how i did it, I used the n-th roots of unity for an equation in the form \[z^n=1\]