one-to-one correspondence

A representation is a morphism (i.e. a function) from a source object to a standard target object.

If A represents B, we have three things: two objects, A and B, which in this context will be sets, and the relation between them, which will be a morphism. When A and B have some additional "structure" (e.g. they are finite sets, or ordered sets, or ...) and we restrict the possible morphisms from A to B to have something to do with that structure.

Example:

Let A be the set {red, blue, yellow, pink}.

Let B be the standard object {1, 2, 3, 4}. Let f be the following rule: red is associated with 1, blue is associated with 2, yellow is associated with 3 and pink is associated with 4. f is a morphism because because this rule is capturing something about the number of elements in A. Therefore we may consider f to be a representation of A.

A group G is a set with a composition defined on pairs of elements such that :

1. There is a neutral element e in G, so that no matter what element in the group is substituted for x. e is sometimes called the identity element.

2. For any element x of G, there is some element y in G so that . That is, every element has an inverse element.

3. For any three elements, x, y, and z in G, we have . This is called the associativity of the composition. Each group has its own law of composition. It can be whatever we define it to be (e.g. addition, multiplication, rotation, ...).

discrete groups

Lie groups

Examples:

Consider the set Z of all the integers (positive, negative and zero). Z is an infinite set. Let the group law be familiar addition. Then Z under this group law is a group.

Consider the set of three rotations of 60, 120 and 180 degrees of an equilateral triangle. The rotations constitute the group law. The elements of the group are the three positions (i.e. orientations) of the triangle.

Groups are used to describe various types of symmetries.

A field is a number system where we can divide by anything nonzero. The integers modulo p (where p is a prime) are a field under addition and multiplication. We say that is a finite field with p elements. We also say that is a number system with characteristic p. Note that does not contain fractions.

is a group under addition.

If we delete the number 0 from , we have a new set of p - 1 elements that is a group under multiplication. The new set is symbolized as "eff-pea-cross" : .

The variety S defined by a Z-equation (or a system of Z-equations) is the function that assigns to any number system A the set of solutions S(A) of the equation.

Many different systems of equations can define the same variety S.

For any positive integer n, let the variety be defined by . Then if n > 2, contains only solutions where one or more of the variables is 0.

It is easier to study . Then some information about (the solution set we are really interested in) can be derived from the preceding three sets using advanced theorems from number theory and algebraic geometry.

Suppose that p and q are odd primes.

1.

2.

3. If , then .

4. If p or q or both , then .

That subset of C which consists of all the equations = 0 where is a polynomial with integer coeficients. It can be shown that is a field. Note that i is an algebraic number (it is the solution to the equation ).

Examples:

All the integers. (they are solutions for the polynomial equation x - n = 0 ).

All rational numbers. (they are solutions for the polynomial equation bx - a = 0 ).

nth roots of every fraction. ( they are solutions for the equation .

It can be shown that the sum, difference, product and quotient of any two algebraic numbers will also be an algebraic number.

p is not in . (This is difficult to prove.)

It can be difficult to prove whether a particular number is or is not in .

There are lots of algebraic numbers and they are connected in many complicated ways.

1. is the set of all complex numbers that can appear as roots of Z-polynomials.

2. The absolute Galois group G is made up of all permutations g of that preserve addition and multiplication.

3. If g is any element of G and f(x) is any Z-polynomial, then as g acts as a permutation of , it permutes the roots of f(x). It never maps some root of f(x) to a nonroot of f(x).

4. G has infinitely many elements.

5. There is only one element of G, other than the neutral element e, for which we can give a complete description. This element is called c, the complex conjugate. No other element of G an be written down explicitly.

6. Any element g of G can be partially described by taking a Z-polynomial f(x), listing its roots and telling what permutation of these algebraic numbers occurs when we apply g.

7. Zorn's Lemma plus some advanced algebra can be used to piece together the partial descriptions of item 6 to get elements of G. That is how we know that G is infinite.

A symmetry is a function that preserves what we feel is important about an object.

Each g in the Galois group G is a symmetry of because it preserves the operations that concern us in algebra, namely, addition and multiplication.