Let F be a field

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 :