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MATH 001 Fields are an important algebraic structure

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MATH 001 Fields are an important algebraic structure

Introduction:

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

Task:

A. Let G be the set of the fifth roots of unity.

1. Use de Moivre’s formula to verify that the fifth roots of unity form a group under complex multiplication, showing all work.

2. Prove that G is isomorphic to Z5 under addition by doing the following:

a. State each step of the proof.

b. Justify each of your steps of the proof.

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.