Let F be a field

ST

ST

ST

ST

ST

a ∈ST

ST

ST

ST

Let F be a field. Let S and T be subfields of F.

1. Use the definitions of a field and a subfield to prove that

is a field, showing all

work.

Since F is a field, and S and T are subfields of F, both S and T are also fields in

and of themselves. They, therefore, have all of the properties of a field.

To show that

is also a field, the following must be shown:

1)

is closed under addition and multiplication:

2)

possesses both additive and multiplicative associativity:

3)

possesses additive and multiplicative commutativity:

4)

possesses distributivity:

5)

possesses an additive identity element and a multiplicative identity

element:

6)

has an additive inverse element and a multiplicative inverse element

for each

: