lundi, novembre 14, 2005

What is a Field? [1]


The structure we shall talk about is known technically as a field. It comprises a set of postulates which apply to a set of objects and to some unspecified operations. In the case of a field, the operations are two in number. Various other structures differ in this regard.
Speaking in a simplified manner, what is necessary to characterise a field is a specified set of objects or elements, and a description of two operations which may be applied to any pair of these elements, and compliance with some few postulates.
Let us see the postulates:

Postulate1: The given set of elements must be closed with respect to the two operations.
Postulate2: The set must contain two identity elements, one with respect to each of the two operations.
Postulate3: For each element in the set, there must also be in the set an inverse element with respect to the first operation.
Postulate4: For each element in the set, except for the addition identity, there must also be in the set an inverse element with respect to the second operation.
Postulate5: Both operations must have the associative and commutative property, and the second operation must be distributive with respect lo the first.

Postulate1: The closure of a given set with respect to a specific operation, for example, addition, implies the following:
If any two elements in the given set are added, the resulting element, or sum, must also be an element of the same set. Likewise with respect to multiplication, if the two elements are multiplied, then the product must also be an element of the given set.
As a quick illustration, suppose a particular given set is the set of the odd numbers. Is this particular set closed with respect to addition? We must ask whether or not the sum of two odd numbers another odd number. If the answer is No, then the set is not closed under addition. On the other hand, what about the product of any two odd numbers? Is it also an odd number? If the answer is Yeas, then the set is closed under multiplication.

Postulate2: What, exactly, is an identity element with respect an operation? We might roughly say that it is a kind of do-nothing element. More specifically, the identity element with respect to addition, for example, is that element which when added to any member of the set causes no change. In other words, the sum of any element and the identity is equal to the original element. In such case, this is precisely the job performed by zero. With respect to multiplication, the identity element is one, for any number multiplied by one remains the same.

Postulates3,4: The inverse of an element x is the element which when used in an operation yields the identity element for that operation. Let us symbolise one operation by a dot and the identity by the letter I. Thus, for any element x we have:

If we apply the same operation over x and the inverse element E
x.E =I

Postulate 5: An operation is associative iff it is independent of the grouping of elements. For instance if, for an operation O, a O (b O c) = (a O b) O c, then O is associative. On the other hand, an operation is deemed commutative if it is independent of the order of the elements.
An operation is distributive in relation to another when it produces the same results when performed on a set of numbers as when performed on members of the set individually. For instance, multiplication is said to be distributive in relation to addition.