# A . Fields

A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multiplication and addition in the real number system (which is itself an example of a field). In each field F there exist unique elements called o and 1 which, under the operations of addition and multiplica­tion, behave with respect to all the other elements of F exactly as their correspondents in the real number system. In two respects, the analogy is not complete: 1) multiplication is not assumed to be commu­tative in every field, and 2) a field may have only a finite number of elements.

More exactly, a field is a set of elements which, under the above mentioned operation of addition, forms an additive abehan group and for which the elements, exclusive of zero, form a multiplicative group and, finally, in which the two group operations are connected by the distributive law. Furthermore, the product of o and any element is de­fined to be o.

If multiplication in the field is commutative, then the field is called a commutative field.