Themes > Science > Mathematics > Calculus > Real Number > Groups and fields

First of all, a group is a mathematical object; it is a triple (X,e,*) with these properties:
• X is a nonempty set.
• e is a specially chosen member of the set X. It is called the identity of the group.
• * is a binary operation on X, which we may call the group operation. This means that whenever p and q are members of X, then p*q is also a member of X.
• (p*q)*r = p*(q*r) for all p,q,r in X.
• p*e = p for every p in X.
• For each p in X, there exists at least one corresponding q in X that satisfies p*q=e. (It can be shown that there is at most one such q, and thus q is uniquely determined by p; we call q the inverse of p.)
The group is said to be abelian (or commutative) if it also satisfies this property:
• p*q = q*p for all p,q in X.

Now, a field is a quintuple (Y,0,+,1,´) with these properties:

• Y is a set, 0 and 1 are two specially chosen members of Y, and + and ´ are two binary operations on Y.
• 0¹1.
• The triple (Y,0,+) is an abelian group.
• The triple (Y\{0},1,´) is an abelian group. (Note that this group has for its set of members, all the members of Y except 0.)
• p´(q+r) = (p´q) + (p´r) for all p,q,r in Y.
(Exercise: A few mathematicians do not include the requirement that 0¹1. Prove that there is only one "field" in which 0=1. For that field, the set Y has only one member.)

Here are a few examples:

• The rational numbers (i.e., numbers like 3/4 and -171/25) are a field.
• The real numbers (i.e., numbers like 87.324116279...) are a field.
• The complex numbers are a field. (Exercise: Verify all the axioms. Also, what is the multiplicative inverse of 3+2i ?)
• The set of all numbers of the form p+qÖ2, where p and q are rational numbers, is a field; it is a subset of the reals and a superset of the rationals. (Exercise: Verify all the axioms. Also, what is the multiplicative inverse of 3+2Ö2 ?)

Following is one more example. We will present a finite field -- that is, a field with only finitely many members. For the set Y, we'll use Y={0,1,2,3,4}. For its addition and multiplication operations, we'll use ordinary addition and multiplication, modified by this rule: If the result of addition or multiplication results in a number greater than 4, subtract 5 or 10 or 15, to get a number in the set Y again. In other words, we'll use these tables for addition and multiplication:

 + . 0 1 2 3 4 . 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3
 ´ . 0 1 2 3 4 . 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1

This field is sometimes called arithmetic modulo 5. (Exercises: Show that a similar field can be given with 5 replaced by any prime number. Show that there is also a field with 4 elements, and a field with 9 elements, but there is no field with exactly 6 elements. Much much harder: It can be shown that there is a field with exactly n elements, for some integer n, if and only if n is of the form pr for some prime number p.)

Information provided by Eric Schechter