FIELD

A GROUP is a set with one operation, and four properties. A group has closure, associativity, an identity. Every element in a group has an inverse. A FIELD is a set (call it F ) with two operations(call them and ). F with operation is a commutative group with identity z F with operation would be a commutative group except that z has no inverse and For every a, b, and c in F: a ( b c ) = ( a b ) ( a c ) This is called the distributive property.

The most familiar example of a FIELD is the set of REAL NUMBERS with operations addition and multiplication. The following example is small but more abstract. Let F = { a, b, c, d, e } The two operations are defined below:

The system has commutativity ab = ba bd = db cd = dc This is a commutative group The system has closure The identity is e etc The inverse of a is d The inverse of b is c The inverse of e is e The system has associativity ( b d ) a = b ( d a ) a a = b e b = b etc

Remove e and this is a commutative group The system has closure The system has commutativity The identity is d The system has associativity ( b c ) a = b ( c a ) d a = b b a = a etc The inverse of a is a The inverse of b is c

a = a etc.