A field is the name given to a pair of numbers and a set of operations which together satisfy several specific laws. A familiar example of a field is the set of rational numbers and the operations addition and multiplication. An example of a set of numbers that is not a field is the set of integers. It is an "integral domain." It is not a field because it lacks multiplicative inverses. Without multiplicative inverses, division may be impossible.
The elements of a field obey the following laws:
- Closure laws: a + b and ab are unique elements in the field.
- Commutative laws: a + b = b + a and ab = ba.
- Associative laws: a + (b + c) = (a + b) + c and a(bc) = (ab)c.
- Identity laws: there exist elements 0 and 1 such that a + 0 = a and a × 1 = a.
- Inverse laws: for every a there exists an element - a such that a + (-a) = 0, and for every a ≠ 0 there exists an element a-1 such that a × a-1 = 1.
- Distributive law: a(b + c) = ab + ac.
Rational numbers (which are numbers that can be expressed as the ratio a/b of an integer a and a natural number b) obey all these laws. They obey closure because the rules for adding and multiplying fractions, a/b + c/d = (ad + cb)/bd and (a/b)(c/d) = (ac)/(bd), convert these operations into adding and multiplying integers which are closed. They are commutative and associative because integers are commutative and associative. The ratio 0/1 is an additive identity, and the ratio 1/1 is a multiplicative identity. The ratios a/b and -a/b are additive inverses, and a/b and b/a (a, b ≠ 0) are multiplicative inverses. The rules for adding and multiplying fractions, together with the distributive law for integers, make the distributive law hold for rational numbers as well. Because the rational numbers obey all the laws, they form a field.
The rational numbers constitute the most widely used field, but there are others. The set of real numbers is a field. The set of complex numbers (numbers of the form a + bi, where a and b are real numbers, and i2 = -1) is also a field.
Although all the fields named above have an infinite number of elements in them, a set with only a finite number of elements can, under the right circumstances, be a field. For example, the set constitutes a field when addition and multiplication are defined by these tables:
With such a small number of elements, one can check that all the laws are obeyed by simply running down all the possibilities. For instance, the symmetry of the tables show that the commutative laws are obeyed. Verifying associativity and distributivity is a little tedious, but it can be done. The identity laws can be verified by looking at the tables. Where things become interesting is in finding inverses, since the addition table has no negative elements in it, and the multiplication table, no fractions. Two additive inverses have to add up to 0. According to the addition table 1 + 1 is 0; so 1, curiously, is its own additive inverse. The multiplication table is less remarkable. Zero never has a multiplicative inverse, and even in ordinary arithmetic, 1 is its own multiplicative inverse, as it is here.
This example is not as outlandish as one might think. If one replaces 0 with "even" and 1 with "odd," the resulting tables are the familiar parity tables for catching mistakes in arithmetic.
One interesting situation arises where an algebraic number such as √ 2 is used. (An algebraic number is one which is the root of a polynomial equation.) If one creates the set of numbers of the form a + b √ 2 , where a and b are rational, this set constitutes a field. Every sum, product, difference, or quotient (except, of course, (a + b √ 2)/0) can be expressed as a number in that form. In fact, when one learns to rationalize the denominator in an expression such as 1/(1 - √ 2 ) that is what is going on. The set of such elements therefore form another field which is called an "algebraic extension" of the original field.
J. Paul Moulton
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McCoy, Neal H. Rings and Ideals. Washington, DC: The Mathematical Association of America, 1948.
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