Exercise 3.22

Suppose the set $R$ consists of four elements: $R=\{0,1,x,y\}$, and two operations $+$ and $\cdot$ are defined on $R$, with the following multiplication tables:

+	0	1	x	y
0	0	1	x	y
1	1	0	y	x
x	x	y	0	1
y	y	x	1	0

$\cdot$	0	1	x	y
0	0	0	0	0
1	0	1	x	y
x	0	x	y	1
y	0	y	1	x
Prove that $R$ is a field. (It would take too long to verify completely the associativity of $+$ and $\cdot$ and the distributivity axiom. They happen to work in this case; please illustrate this fact by choosing an example for each of these properties. On the other hand, make sure you give a complete verification for the other axioms needed to prove that $R$ is a field.)

Answer

The addition and multiplication are clearly closed based on the table.

1. Commutativity of addition. Since the table is symmetric, it is commutative.
2. Additive identity. 0 is the additive identity, can be seen from the first row and column.
3. Associativity of addition. (0 + 1) + x = 1 + x = y = 0 + y = 0 + (1 + x)
4. Additive inverse. Each element has an additive inverse since each row has a 0.
5. Commutativity of multiplication. Since the table is symmetric, it is commutative.
6. Associativity of multiplication. 1(xy) = 1(1) = 1 = xy = (1x)y
7. Multiplicative identity. 1 is the multiplicative identity, can be seen from the second row and column.
8. Distributivity. x(y + y) = x0 = 0 = 1 + 1 = xy + xy
9. Multiplicative inverse. Each nonzero element has a multiplicative inverse since each nonzero row has a 1.