Zeta functions. No.2

Yoshifumi Tsuchimoto

In this lecture we define and observe some properties of conguent zeta functions.

LEMMA 2.1   For any prime number $p$ , $\mathbb{Z}/p \mathbb{Z}$ is a field. (We denote it by $\mathbb{F}_p$ .)

Funny things about this field are:

LEMMA 2.2   Let $p$ be a prime number. Let $R$ be a commutative ring which contains $\mathbb{F}_p$ as a subring. Then we have the following facts.

1. $$\underbrace{1+1+\dots+1 }_{\text{$p$-times}}=0$$
   holds in $R$ .
2. For any $x,y\in R$ , we have
   $$(x+y)^p=x^p +y^p$$

We would like to show existence of ``finite fields''. A first thing to do is to know their basic properties.

LEMMA 2.3   Let $F$ be a finite field (that means, a field which has only a finite number of elements.) Then:

1. There exists a prime number $p$ such that $p=0$ holds in $F$ .
2. $F$ contains $\mathbb{F}_p$ as a subfield.
3. $q=\char93 (F)$ is a power of $p$ .
4. For any $x\in F$ , we have $x^q-x=0$ .
5. The multiplicative group $(F_q)^{\times}$ is a cyclic group of order $q-1$ .

The next task is to construct such field. An important tool is the following lemma.

LEMMA 2.4   For any field $K$ and for any non zero polynomial $f\in K[X]$ , there exists a field $L$ containing $L$ such that $f$ is decomposed into linear factors in $L$ .

To prove it we use the following lemma.

LEMMA 2.5   For any field $K$ and for any irreducible polynomial $f\in K[X]$ of degree $d>0$ , we have the following.

1. $L=K[X]/(f(X))$ is a field.
2. Let $a$ be the class of $X$ in $L$ . Then $a$ satisfies $f(a)=0$ .

Then we have the following lemma.

LEMMA 2.6   Let $p$ be a prime number. Let $q=p^r$ be a power of $p$ . Let $L$ be a field extension of $\mathbb{F}_p$ such that $X^q-X$ is decomposed into polynomials of degree $1$ in $L$ . Then

1. $$L_1=\{x \in L; x^q=x\}$$
   is a subfield of $L$ containing $\mathbb{F}_p$ .
2. $L_1$ has exactly $q$ elements.

Finally we have the following lemma.

LEMMA 2.7   Let $p$ be a prime number. Let $r$ be a positive integer. Let $q=p^r$ . Then we have the following facts.

1. There exists a field which has exactly $q$ elements.
2. There exists an irreducible polynomial $f$ of degree $r$ over $\mathbb{F}_p$ .
3. $X^q-X$ is divisible by $f$ .
4. For any field $K$ which has exactly $q$ -elements, there exists an element $a\in K$ such that $f(a)=0$ .

In conclusion, we obtain:

THEOREM 2.8   For any power $q$ of $p$ , there exists a field which has exactly $q$ elements. It is unique up to an isomorphism. (We denote it by $\mathbb{F}_q$ .)

The relation between various $\mathbb{F}_q$ 's is described in the following lemma.

LEMMA 2.9   There exists a homomorphism from $\mathbb{F}_q$ to $\mathbb{F}_{q'}$ if and only if $q'$ is a power of $q$ .

EXERCISE 2.1   Compute the inverse of $113$ in the field $\mathbb{F}_{359}$ .