I am reading Koblitz p-adic analysis book and I am on page 111. The lemma is that $\zeta_{H_f}(T)$ has coefficients in $\mathbb{Z}$. I could follow the rest of the book from page 1 just fine until the proof of this lemma which is horrendously written, probably the worst written proof I seen this month.
Anyway, start with a finite field $\mathbb{F}_q$ and a polynomial $f\in F_{q}[X_1,...,X_n]$. We define $H_f = \{ x\in \mathbb{A}^n(\mathbb{F}_q) : f(x) = 0\}$. For a finite extension $K$ we define $H_f(K) = \{x\in \mathbb{A}^n(K): f(x) = 0\}$. We define numbers $N_s$ to be the size of the set $|H_f(K)|$ where $K$ is the finite extension of degree $s$. We define the zeta function over this hypersurface to be, $$ \zeta_{H_f} (T) = \exp \left( \sum_{s=1}^{\infty} \frac{N_s T^s}{s} \right) $$
The first line of the proof says, "We consider the $K$-points $P=(x_1,...,x_n)$ of $H_f$ ($K$ a finite extension of $\mathbb{F}_q$) according to the least $s=s_0$ for which all $x_i \in \mathbb{F}_{q^{s_0}}$."
What does this even mean?
