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.) |