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.

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