We will start by examining some of the consequences of making a finiteness assumption on the structures we examine. One of the major changes that this assumption permits is that we can now count things. This opens up the possibility of using new tools and providing different methods for proving things.

**Theorem:** *The characteristic of a field, if not 0, is a prime number*.

The set of sums of 1 forms a ring contained in * F*, and the quotient field of that ring is the smallest field contained in

**Theorem:** *A finite field of characteristic p has p ^{n} elements for some positive integer n.*

As any field can be viewed as a vector space over any of its subfields, we can think of * F* as a vector space over its prime subfield. Since

**Theorem:** *For each prime p and positive integer n, there exists a finite field with p ^{n} elements.*

This theorem is proved by construction. The details of such a construction can be found on the finite fields web page.

The number of elements in a finite field is called its *order*.

**Theorem:** *All finite fields of the same order are isomorphic.*

This theorem permits us to unambiguously refer to the finite field of order q, denoted by **GF**(q), where q = p^{n} for some prime p.

On of the most important structural features of a finite field is:

**Theorem:** *The multiplicative group GF(q)^{*} = GF(q)-{0} is a cyclic group*.

A generator of **GF**(p^{n})* is called a **primitive element** of
**GF**(p^{n}).

**Theorem:** **GF**(p^{m} ) is a subfield of ** GF**(p^{n} ) iff m | n.

An *automorphism* of a field is a bijection of the field onto itself which is both an additive and multiplicative homomorphism (i.e., preserves both addition and multiplication). For a finite field **GF**(q) of characteristic p, the map f(x) = x^{p} is an automorphism, called the *Frobenius automorphism*.

**Theorem:** *The automorphism group of GF(q) of characteristic p is cyclic and generated by the Frobenius automorphism.*

**Theorem:** *A rank n vector space over GF(q) has q ^{n} vectors.*

Recall the important rank formula for subspaces U and W of the vector space V:

**Theorem:** *The vector space V(n+1,q) has (q ^{n+1}-1)/(q-1) = q^{n} + q^{n-1} + ... + q + 1 rank 1 subspaces.*

Given the vector space V(n+1,q), we define PG(n,q) as follows:

The objects of PG(n,q) consist of:

*points*, which are the rank 1 subspaces of V(n+1,q).*lines*, which are the rank 2 subspaces of V(n+1,q).*planes*, which are the rank 3 subspaces of V(n+1,q).- ...
*i-spaces*, which are the rank i+1 subspaces of V(n+1,q).- ...
*hyperplanes*, which are the rank n subspaces of V(n+1,q).

We can now rephrase statements about vector spaces in terms of the geometric objects of the projective geometry. For instance, the statement in the previous section about two distinct rank 1 subspaces becomes two distinct points determine a unique line. The statements about rank 2 subspaces become, in PG(2,q) every two distinct lines meet at a unique point, while in higher dimensional projective spaces two distinct lines which meet lie in a unique plane and if they do not meet (are *skew*) lie in a unique 3-space (*solid*).

Example

Coordinates

Restriction to dimensions 2 and 3

Degenerate Sets

Ovals

Conics

Cones