1 - Fields

Summary

In this chapter the basic algebraic objects of a group, a ring and a field are defined. It is shown that a finite field has q elements, where q is a prime power, and that there is a unique field with q elements. We define an automorphism of a field and introduce the associated trace and norm functions. Some lemmas related to these functions are proven in the case that the field is finite. Finally, some additional results on fields are proven which will be needed in the subsequent chapters.

Rings and fields

A group G is a set with a binary operation ◦ which is associative ((a ◦ b) ◦ c = a ◦ (b ◦ c)), has an identity element e (a ◦ e = e ◦ a = a) and for which every element of G has an inverse (for all a, there is a b such that a ◦ b = b ◦ a = e). A group is abelian if the binary operation is commutative (a ◦ b = b ◦ a).

A commutative ring R is a set with two binary operations, addition and multiplication, such that it is an abelian group with respect to addition with identity element 0, and multiplication is commutative, associative and distributive (a(b + c) = ab + ac) and has an identity element 1.

The set of integers ℤ is an example of a commutative ring.

An ideal a of a ring R is an additive subgroup with the property that ra ∈ a for all r ∈ R and a ∈ a. For example, the multiples of an element r ∈ R form an ideal, which is denoted by (r).

A coset of a is a set r + a = {r + a | a ∈ a}, for some r ∈ R.