Cyclotomic cosets and minimal polynomials
0
Let $mathbb{F}_{p^m}$ be a field and let $alpha in mathbb{F}_{p^m}$ . Let $M^{(i)}$ be the minimal polynomial of $alpha^i$ . Then I know that $M^{(i)}(x) = prod_{j in C_s} (x - alpha^j)$ , where $i$ is in the cyclotomic coset $C_s$ . Further, from this, we get that $x^{p^m - 1} - 1 = prod_{s} M^{s}(x)$ , where $s$ runs through the coset representatives mod $p^m - 1$ How do we get this last equation? Is $alpha$ assumed to be primitive here?
field-theory finite-fields minimal-polynomials
share | cite | improve this question
asked Nov 26 '18 at 20:03
the man
697 7 15
...
