I have been trying a long time figure out how to state the following congruent
show $x^p−x\equiv x(x−1)(x−2)⋯(x−(p−1)) \pmod p$
I got some hint: find all the roots of $x^p − x$ using Fermat's theorem, write x^p − x as a product of its factors and compare the coefficient of the leading term of both sides.
I know that by Fermat's Little theorem that $x^p -x$ could be factory into
$$x(x -1)(x -2)...(x -p+1)$$ and the coefficient of the leading term is $1$, but I dont know what to do next.
I do know by Wilson's theorem If p is prime then x^p−x≡x(x−1)(x−2)⋯(x−(p−1))≡xp+⋯+(p−1)!x(modp). but I dont know what to do with this got it from