Question

Find all seventh roots of unity.

Answer 1

`1=e^(i0)=e^(i2\pi)=e^(i4\pi)=\cdots`

`\implies1^(1/7)=e^(i(0+2\pi k)/7)`

This means the seventh roots of unity are


`e^(i0),e^(i2\pi/7),e^(i4\pi/7),e^(i6\pi/7),e^(i8\pi/7),e^(i10\pi/7),e^(i12\pi/7),e^(i14\pi/7),\ldots`

but notice that
`\frac{14\pi}7=2\pi\equiv0\mod{2\pi}`, which means the seventh roots begin to repeat, so we only need to worry about
`k=0,1,\ldots,6`.