**Yoshifumi Tsuchimoto**

Funny things about this field are:

- For any
, we have

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

- There exists a prime number such that holds in .
- contains as a subfield.
- is a power of .
- For any , we have .
- The multiplicative group is a cyclic group of order .

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

To prove it we use the following lemma.

- is a field.
- Let be the class of in . Then satisfies .

Then we have the following lemma.

- has exactly elements.

Finally we have the following lemma.

- There exists a field which has exactly elements.
- There exists an irreducible polynomial of degree over .
- is divisible by .
- For any field which has exactly -elements, there exists an element such that .

In conclusion, we obtain:

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