Hi! I just have a short question: I want to proof the following claim: Let K be a field with characteristic p>0 and f be a polynomial in Z[X] (with Z being the set of integers). Then f(y^p)=f(y)^p for any y in K.
My consideration is this: f=a_0 + a_1 x + a_2 x^2 + ... + a_n x^n Using the binomial theorem it easily follows that f(y)^p=(a_0)^p + (a_1 y)^p + (a_2 y^2)^p + ... + (a_n y^n)^p = a_0 + a_1 y^p + ... + a_n (y^p)^n = f(y^p), as (a_i)^p=a_i. But this last implication only holds if the coefficients are in Z/pZ which is not demanded.
So my questions: Is there a mistake in the description of the exercise? Or are the coefficients to be treated modulo p; if yes, why? Thankful for any tips, Banach!