Field Properties

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Field Properties

David Hilbert, a famous German mathematician (1862–1943), called mathematics the rules of a game played with meaningless marks on paper. In defining the rules of the game called mathematics, mathematicians have organized numbers into various sets, or structures, in which all the numbers satisfy a particular group of rules. Mathematicians call any set of numbers that satisfies the following properties a field : closure, commutativity, associativity, distributivity, identity elements, and inverses.

Determining a Field

Consider the set of non-negative even numbers: {0, 2, 4, 6, 8, 10, 12,…}. To determine whether this set is a field, test to see if it satisfies each of the six field properties.

Closure. When any two numbers from this set are added, is the result always a number from this set? Yes, adding two non-negative even numbers will always result in a non-negative even number. The set of non-negative even numbers is therefore closed under addition.

Is the set of even non-negative numbers also closed under multiplication? Yes, multiplying two non-negative even numbers will also always result in a non-negative even number. The closure property applies to the set of non-negative even numbers under the two operations of addition and multiplication.

Commutativity. Notice also that, with any two numbers from this set (a, b ), a + b = b + a and ab = ba. Therefore, the commutative property for addition and for multiplication applies also.

Associativity. It is also true that (a + b ) + c = a + (b + c ) and (ab )c = a (bc ). The associative property for addition and for multiplication thus applies for the set of non-negative even numbers.

Distributivity. If the distributive property applies to the set of non-negative even numbers, a (b + c )= ab + ac. Since this is true for any non-negative even numbers, the set does satisfy this property.

Identity Elements. Within this set of non-negative even numbers, is there an identity element for addition? That is, is there a number n such that adding that number leaves a non-negative even number unchanged in value? Does the set contain an n such that a + n = a ? Yes, n can be 0, so 0 is the identity element for addition in this set.

Is there a corresponding identity element for multiplication in this set? No. Here the set of non-negative even numbers fails the test. There is no number p in this set such that ap = a. The number p could be 1 because 1 is an identity element for multiplication, but 1 is not in the set of nonnegative even numbers.

Because the identity property is not satisfied by the set of non-negative even numbers, the set does not form a field.

Inverses. The set of non-negative even numbers also does not satisfy the sixth property for a field. This set does not contain additive and multiplicative inverses for each number in the set. An additive inverse for 2 might be −2, since 2 + (−2) = 0, but −2 is not in this set. A multiplicative inverse for 2 might be ½, since 2 (½) = 1, but ½ is also not in this set.

Numbers Sets that Are Fields

Are there sets of numbers that are fields—that is, that satisfy all six of the field properties—closure, commutativity, associativity, distributivity, identity elements, and inverses? If the set of non-negative even integers is expanded to include the negative integers (to supply the additive inverses), all the integers (so that 1 is the multiplicative identity), and all the rational numbers (such as ½, to supply all the multiplicative inverses, or reciprocals), then the result is the set of all rational numbers .

The set of rational numbers is a field because it satisfies all six properties. This set is closed because adding or multiplying any two rational numbers results in a rational number. It is commutative, associative, and distributive. It contains an additive identity, 0, and a multiplicative identity, 1. Every number in the set (except 0) has an additive inverse and a multiplicative inverse in the set.

Notice that the rules for a field do not require that 0 have a reciprocal; division by 0 is undefined.

Another set of numbers that form a field, because they satisfy all six of the field properties, is the set of all numbers on the real number line. This set of all real numbers is formed by joining the rational numbers to all the irrational numbers. Recall that an irrational number cannot be expressed as the ratio of two integers.

A third set of numbers that forms a field is the set of complex numbers. Complex numbers are all the numbers that can be written in the form abi where a and b are real numbers, and i is the square root of -1.

There are other sets of numbers that form a field. For example, consider this set of numbers: {0, 1, 2, 3}. The operation of addition is defined in the following way. Add the two numbers in the set and, if the result is 4 or more, subtract the number 4 until a number, called the sum, remains that is in the set. This method of arithmetic is called modular arithmetic (in this case, mod 4).

Thus, 2 + 3, for example, yields 1 (since 5 − 4 = 1); and 1 + 3 yields 0 (since 4 − 4 = 0). When addition is defined in this way (in this case, as mod 4), then this set is closed under addition. The identity element for addition is 4 because, for example, 2 + 4 yields 2, so that adding 4 leaves a number from this set unchanged.

Notice that each number in the set (other than 0) does have a multiplicative inverse, since, using mod 4 arithmetic, 1 × 1 = 1, and 2 × 3 = 2, and 3 × 1 = 3. The set is also closed under multiplication, using mod 4 arithmetic. It also shows associativity, commutativity, and distributivity under these definitions of addition and multiplication. There are many other finite sets that are finite fields when this kind of modular arithmetic is used.