# 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 , is a field. (We denote it by .)

LEMMA 2.2   Let be a prime number. Let be a commutative ring which contains as a subring. Then we have the following facts.
1. holds in .
2. For any , we have

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

LEMMA 2.3   Let be a finite field (that means, a field which has only a finite number of elements.) Then:
1. There exists a prime number such that holds in .
2. contains as a subfield.
3. is a power of .
4. For any , we have .
5. The multiplicative group is a cyclic group of order .

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

LEMMA 2.4   For any field and for any non zero polynomial , there exists a field containing such that is decomposed into linear factors in .

To prove it we use the following lemma.

LEMMA 2.5   For any field and for any irreducible polynomial of degree , we have the following.
1. is a field.
2. Let be the class of in . Then satisfies .

Then we have the following lemma.

LEMMA 2.6   Let be a prime number. Let be a power of . Let be a field extension of such that is decomposed into polynomials of degree in . Then
1. is a subfield of containing .
2. has exactly elements.

Finally we have the following lemma.

LEMMA 2.7   Let be a prime number. Let be a positive integer. Let . Then we have the following facts.
1. There exists a field which has exactly elements.
2. There exists an irreducible polynomial of degree over .
3. is divisible by .
4. For any field which has exactly -elements, there exists an element such that .

In conclusion, we obtain:

THEOREM 2.8   For any power of , there exists a field which has exactly elements. It is unique up to an isomorphism. (We denote it by .)

The relation between various 's is described in the following lemma.

LEMMA 2.9   There exists a homomorphism from to if and only if is a power of .

EXERCISE 2.1   Compute the inverse of in the field .