Congruent zeta functions. No.1

Yoshifumi Tsuchimoto

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

For a good brief account of finite fields, consult Chapter I of a book [1] of Serre.

LEMMA 1.1   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 1.2   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 1.3   Let $F$ be a finite field (that means, a field which has only a finite number of elements.) Then we have,

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 1.4   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 polynomials of degree $1$.

To prove it we use the following lemma.

LEMMA 1.5   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 1.6   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 1.7   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 1.8   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 1.9   There exists a homomorphism from $\mathbb{F}_q$ to $\mathbb{F}_{q'}$ if and only if $q'$ is a power of $q$.

EXERCISE 1.1   Compute the inverse of $113$ in the field $\mathbb{F}_{359}$.