1. FIELD

2. 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.

3. 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:

4. 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

5. 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

6. a = a
etc.