EXAMPLES OF FIELDS:

• The complex numbers C, under the usual operations of addition and multiplication.
  

• The rational numbers Q = { a/b | a, b in Z, b ≠ 0 } where Z is the set of integers.
  

• The real numbers R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered field ; it is this structure which provides the foundation for most formal treatments of calculus.
  

• For a given prime number p, the set of integers modulo p is a finite field with p elements:
  

Integers, polynomial rings, and matrix rings are NOT fields

The integers Z are a commutative ring but are NOT a field because the only elements in Z that have multiplicative inverses are 1 and -1

It is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is an integer, but the only solution to the equation ab = 1 in this case is b = 1/2. We cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails)

Since not every element of (Z,·) has an inverse, (Z,·) is not a group.

An algebraic field is, by definition, a set of elements (numbers) that is closed under the ordinary arithmetical operations of addition, subtraction, multiplication, and division (except for division by zero). For example, the set of rational numbers is a field, whereas the integers are not a field, because they are not closed under the operation of division (i.e., the result of dividing one integer by another is not necessarily an integer). Why are you dividing? I thought addition and subtraction were the only two operations considered when categorizing groups. The real numbers also constitute a field, as do the complex numbers.

The integers are not a field (no inverse).

Z, the integers, are not a field. There is no multiplicative inverse for any elements other than ±1. That is, there is no element y for which 2y = 1 in the integers.

Note that the natural numbers are not a field, as M3 is generally not satified, i.e. not every natural number has an inverse that is also a natural number.

Algebraic Sturctures

What is a Field?

A B S R A C T   A L G E B R A

Lesson #1

Before tackling the concept of a field, make sure that you have a thorough understanding of groups and rings. You will recall that a ring is a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), provided that the set is an abelian group under addition, that multiplication is associative, and that both the left distributive law and the right distributive law hold for all elements of the set.

And don’t forget that multiplication need not be commutative for a group to be a ring, but if it is, we say that they ring is commutative.

Remember also that a ring need not have an identity under multiplication, but if a ring other than {0} does have identity under multiplication, we say that the ring has identity or unity.

And finally, recall that a nonzero element of a commutative ring with unity need not have a multiplicative inverse, but when it does, we say that the nonzero element is a unit of the ring.

Don’t worry. We are going someplace with all this! Specifically, it brings us to a field, which is simply a commutative ring with a unity in which every nonzero element is a unit.

A field is a commutative ring with unity in which every nonzero element is a unit.

Memorize it! (To define a field informally, we might say that it is a ring that is commutative, and has both identity and inverses, all under multiplication.)

REVIEW

• A group is a nonempty set, together with a binary operation (usually called multiplication) that assigns to each ordered pair of elements (a,b) some element from the same set, denoted by ab. (The set is a group under the given binary operation if and only if the properties of closure, associativity, identity, and inverses are satisfied.)
  

• A ring is a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), provided that the set is an abelian group under addition, that multiplication is associative, and that both the left distributive law and the right distributive law hold for all elements of the set. So, a ring is in Abelian group under addition, also having an associative multiplication that is left and right distributive over addition.
  

• A field is a commutative ring with unity in which every nonzero element is a unit.
  

MODULO

(This usage is from Gauss's book.) Given the integers a, b and n, the expression a ≡ b (mod n) (pronounced "a is congruent to b modulo n") means that a and b have the same remainder when divided by n, or equivalently, that a − b is a multiple of n. For more details, see modular arithmetic.