- 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

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).

What is a Field?

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*.

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.)

- 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.

(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.