Geometry Seminar Lecture Notes 1

Review

For the purposes of this review we will assume that you are familiar with the definition of a field, the definition of a vector space over a field and the fact that the integers mod p, for p a prime number, form a field, which will be denoted GF(p) throughout these lecture notes (The GF stands for Galois Field.)

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.

Finite Fields

First, let us recall the definition of the characteristic of a (arbitrary) field. Let our field be denoted by F and let 1 be the multiplicative identity in F (note that 1 is not the additive identity which is denoted by 0). Since F is closed under addition, adding 1 to itself any number of times always results in an element of the field. Thus, 1, 2 = 1 + 1, 3 = 1 + 1 + 1, ..., etc. are all elements of F. There are two possibilities, either there is a first time that this sequence of sums of 1 equals 0 (i.e., n = 1 + 1 + ... + 1 = 0) or this never happens. In case that this never happens, we say that the field has characteristic 0, otherwise it has characteristic n. For a finite field, the sequence of sums can not be infinite, thus it must have repetitions and it is then easy to show that some sum must be 0. Thus, finite fields always have non-zero characteristic.

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 F (a subfield of F). For fields of characteristic 0 this smallest subfield is isomorphic to Q, the field of rational numbers, while for fields of characteristic p it is isomorphic to GF(p). For finite fields, this smallest subfield is called the prime subfield.

Theorem: A finite field of characteristic p has pn 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 F is finite, this vector space must have some finite dimension, say n. The theorem follows by counting the number of vectors in this vector space.

Theorem: For each prime p and positive integer n, there exists a finite field with pn 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 = pn 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(pn)* is called a primitive element of GF(pn).

Theorem: GF(pm ) is a subfield of GF(pn ) 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) = xp 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.

Vector Spaces over Finite Fields

We are interested only in vector spaces of finite dimension. To avoid a notational difficulty that will become apparent later, we will use the word rank (or algebraic dimension) for the dimension (number of vectors in any basis) of the vector space.

Theorem: A rank n vector space over GF(q) has qn vectors.

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

rank(<U,W>) = rank(U) + rank(W) - rank(UintersectW)
Let V = V(n,q) denote a rank n vector space over GF(q). A rank 1 subspace of V consists of all the scalar multiples of a given vector, thus there are q vectors in such a subspace (including the zero vector). By the rank formula, the join of any two distinct rank 1 subspaces has rank 2, since they can only intersect in the zero vector which as a subspace has rank 0. If we examine two distinct rank 2 subspaces, U and W, we notice that there are several possibilities for the rank of their join. If n = 3 then rank(<U,W>) = 3 and the rank formula says that they must intersect in a rank 1 subspace. If n > 3, then there are two possibilities, either the join has rank 4 (when their intersection is just the zero vector) or rank 3 (if they intersect in a rank 1 subspace.)

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

Projective Geometries

A projective geometry is a geometric structure consisting of various types of objects (points, lines, planes, etc.) and the relations between them which satisfies a set of axioms. Here, we will not develop the subject axiomatically (as is done in M6221) but will settle for an algebraic construction starting with a vector space which will give a structure that satisfies the (unstated) axioms. We will start with the vector space V(n+1,q) and construct the geometric structure PG(n,q), called the projective geometry of dimension n over GF(q). The word "dimension" is used here in the classical geometric sense in which lines have 1 dimension, planes have 2 dimensions, etc. This use of the term is different from (but related to) the algebraic dimension of vector spaces (rank). Since in this treatment both geometries and vector spaces appear together, it is inevitable that confusion will arise unless one is very careful. We shall always use the term dimension in its geometric sense, sometimes using projective dimension for additional emphasis.

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

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

The relationship between the objects of PG(n,q) is called incidence and is defined by containment of the corresponding subspaces. The incidence relation is meant to be symmetric, so we say that a point is incident with a line (the point is on the line) or that a line is incident with a point (the line passes through the point) if the rank 1 subspace is contained in the rank 2 subspace.

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

Quadratic Sets

Definition

Degenerate Sets

Ovals

Conics

Cones