On page 5 of Serre's book "A course in Arithmetic", he gives the proof of Chevalley-Warning Theorem. Note that $K$ is a field with $q$ elements where $q=p^n$ and $p$ a prime number.
Theorem (Chevalley-Warning): Let $f_\alpha \in K[X_1,...X_n]$ be polynomials in $n$ variables such that $\sum_{\alpha} deg(f_\alpha)<n$, and let $V$ be the set of their common zeros in $K^n$. One has Card($V$)$\equiv 0$ (mod $p$).
Proof: Put $P=\prod_\alpha (1-f_\alpha^{q-1})$ and let $x\in K$. Then after some discussion this $P$ is actually a characterstic function of $V$($P(x)=1$ if $x\in V$ and 0 otherwise). Then for every polynomial $f$, define $S(f)=\sum_{x\in K^n} f(x)$. And here my question comes when he claims Card($V$)$\equiv S(P)$ (mod $p$). Isn't it just a equality as the function $S(P)$ runs over all elements in $K^n$ and gives $1$ if $x\in V$?
Thank you so much!
Note that $S(P)\in K$ which is a field of characteristic $p$.