Fields are an important algebraic structure, and complex numbers have that structure.

A. Use de Moivre’s formula to verify that the 5th roots of unity form a group under complex multiplication, showing all work.
B. 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 S ∩ T is a field, showing all work.

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Let S and T be subfields of F.
We first show that the intersection of S and T is non-empty. Let 0 denote the identity element with respect to addition in F. Since S and T are both subfields of F, each must contain 0, and therefore 0 is in the intersection.

We next show that the intersection is a commutative ring.
First the intersection is closed under addition + and multiplication...
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