## A fact about multivariate polynomials over ZZ

Let $f(x_1,\ldots,x_n) \in \mathbb{Z}[x_1,\ldots,x_n]$, and let $p$ be a prime. Prove that $f(x_1,\ldots,x_n)^p = f(x_1^p,\ldots,x_n^p)$ mod $p$.

Remember that $\alpha^p \equiv \alpha$ mod $p$ for all $\alpha$. Say $f = \sum_i c_i \prod_j x_j^{e_{i,j}}$.

Then $f(x_1,\ldots,x_n)^p = (\sum_i c_i \prod_j x_j^{e_{i,j}})^p$ $= \sum_i c_i^p \prod_j x_j^{e_{i,j}p}$ $= \sum_i c_i \prod_j (x_j^p)^{e_{i,j}}$ $= f(x_1^p,\ldots,x_n^p)$ as desired.