Consider the set S ab with two elements Endow S with an add

Consider the set S = {a,b} with two elements. Endow S with an addition operation +:S x S implies S, as well as a multiplication operation
*: S x S implies S, as defined in the chart below.

a) Verify that S is a field.

b) Identify the elements of S that are 0, 1 and -1

Solution

a) A field is a commutatative ring with unity in wicha all nonzero elements have multiplicative inverse

Clearly S is closed under addition and multiplication

+ is associative , commutative , the additive identity , ie 0 of S is y and -x = x and -y = y

Thus S is an additive abelian group

Now * satisfies associative and both the distriutive laws

Thus S is a ring

Now * is commutative , sothat S is a commutative ring and unity in S is x and the nonzero element x of S has multiplicative inverse x

Thus S is a field

b) 0= y

1=x

-1 = x