Let p be a prime number. Find all roots of x^(p-1) in Z_p
I have this definition.
Let f(x) be in F[x]. An element c in F is said to be a root of multiplicity m>=1 of f(x) if (x-c)^m|f(x), but (x-c)^(m+1) does not divide f(x).
I'm not sure if I use this idea somehow or not.
The reason is that is a field. As a field has no zero divisors, then you are immediately done. If you have no idea what a field is, then continue reading...
If then , as is prime. Therefore, there exists such that (using the Euclidean algorithm, etc.) That is, there exists such that ( is just ).
So, if then , a contradiction.