First properties of congruent Zeta function

Let us first recall an elementary formula

LEMMA 4.2  

$$\sum_{k=1}^\infty \frac{1}{k} T^k = -\log(1-T)$$

DEFINITION 4.3   Let $\mathbbm{k}$ be a ring. We define $\mathbb{A}^n$ as the affine spectrum of the polynomial ring $\mathbbm{k}[X_1,\dots,X_n]$. For any field (or ring) $L$ over $\mathbbm{k}$, we have

$$\mathbb{A}^n(L)=\{ (x_1,x_2,\dots,x_n); x_1,x_2,\dots, x_n\in L\}.$$

PROPOSITION 4.4  

$$Z(\mathbb{A}^n/\mathbb{F}_q,T)= \frac{1}{1-q^n T}$$

PROPOSITION 4.5   Let $V,W,W_1,W_2$ be affine varieties.

1. If $\char93 V(\mathbb{F}_{q^s})= \char93 W(\mathbb{F}_{q^s})$ for any $s$ ,then $Z(V/\mathbb{F}_q,T) =Z(W/\mathbb{F}_q,T)$.
2. If $\char93 V(\mathbb{F}_{q^s})= \char93 W_1(\mathbb{F}_{q^s})+\char93 W_2(\mathbb{F}_{q^s})$ for any $s$ ,then:
   $$Z(V/\mathbb{F}_q,T) =Z(W_1/\mathbb{F}_q,T) Z(W_2/\mathbb{F}_q,T).$$

PROPOSITION 4.6   Let $f\in \mathbb{F}_q[X]$ be an irreducible polynomial in one variable of degree $d$. Let us consider $V=\operatorname{Spec}(\mathbbm{k}[X]/(f)$. Then:

1. $$% latex2html id marker 884V(\mathbb{F}_{q^s})= \begin{cases} d & \text {if } d \vert s \\ 0 & \text {otherwise} \end{cases}$$

2. $$Z(V/\mathbb{F}_q,T) = \frac{1}{1-T^d}$$

EXERCISE 4.3   Describe what happens when we omit the assumption of $f$ being irreducible in Proposition 4.6.