Corollary 5 The conjugate of a unit root is also a unit root.
Proof From the property of complex numbers
${\displaystyle z\cdot {\overline {z}}=|z|^{2}} {\displaystyle z\cdot {\overline {z}}=|z|^{2}}\:\text{ and }\:{\displaystyle |\epsilon _{k}|=1} {\displaystyle |\epsilon _{k}|=1}, {\displaystyle {\overline {\epsilon _{k}}}={\frac {|\epsilon _{k}|^{2}}{\epsilon _{k}}}={\frac {1}{\epsilon _{k}}}=\epsilon _{-k}=\epsilon _{n-k}} {\displaystyle {\overline {\epsilon _{k}}}={\frac {|\epsilon _{k}|^{2}}{\epsilon _{k}}}={\frac {1}{\epsilon _{k}}}=\epsilon _{-k}=\epsilon _{n-k}}$
Can someone explain how $\epsilon _{-k}=\epsilon _{n-k}$
Answer 1: "$n-k$": go all the way around the unit circle, then back up by $k$. "$-k$" : just back up by $k$.
Answer 2: $\epsilon_{n-k} = \epsilon_n \epsilon_{-k} = 1 \epsilon_{-k} = \epsilon_{-k}$.
These two answers say exactly the same thing.