In a regular Field $F$. If $p$ is a prime number, all $p$th roots of units (roots of the polynomial $x^p - 1_F$), expect $1_F$, are primitive?
I know this is true for $F=\mathbb C$, but is it true for any field $F$?
If it is, how do i prove it?
Solution 1:
Yes it is true, assuming that all $p$ roots of unity are in the field $F$ that is.
HINT: The $p$ $p$-th roots of unity form a group under multiplication. If this--your claim that is--were not true, then there would be a proper subgroup of a $p$-elrment group, the subgroup of cardinality greater than $1$.