In the book "Fundamentals of Error-Correcting Codes" this was written
If $t =$ ord$_n(q)$, then $F_{q^t}$ contains a primitive nth root of unity $\alpha$, but no smaller extension field of $F_q$ contains such a primitive root. (As $\alpha^i$ are distinct for $0 < i < n$ and $(\alpha^i)^n = 1$); $F_{q^t}$ contains all the roots of $x^n-1$. So $F_{q^t}$ is called a splitting field of $x^n-1$ over $F_q$.
could any one help me
How this (As $\alpha^i$ are distinct for $0 \le i < n$ and $(\alpha^i)^n = 1$) ?
the book page 122 \
update :$\alpha^i$ not $\alpha_i$