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Hello! I am really struggling...

Hello! I am really struggling in my course and I was looking to get some help on the following questions...

(Also, I noticed that you have already answered part of the question here... http://www.justanswer.com/pre-calculus/8p4yg-fields-important-algebraic-structure-complex-numbers.html)

A. Use the definition for a ring to prove that Z7 is a ring under the operations + and × defined as follows:

[a]7 +**7 = [a + b]7 and [a]7 × ****7 = [a × b]7**

Note: On the right-hand-side of these equations, + and × are the usual operations on the integers, so the modular versions of addition and multiplication inherit many properties from integer addition and multiplication.

1. State each step of your proof.

2. Provide written justification for each step of your proof.

B. Use the definition for an integral domain to prove that Z7 is an integral domain.

1. State each step of your proof.

2. Provide written justification for each step of your proof.

C. 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.

D. 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.

I look forward to hearing back from you. Thank you!

(Also, I noticed that you have already answered part of the question here... http://www.justanswer.com/pre-calculus/8p4yg-fields-important-algebraic-structure-complex-numbers.html)

A. Use the definition for a ring to prove that Z7 is a ring under the operations + and × defined as follows:

[a]7 +

Note: On the right-hand-side of these equations, + and × are the usual operations on the integers, so the modular versions of addition and multiplication inherit many properties from integer addition and multiplication.

1. State each step of your proof.

2. Provide written justification for each step of your proof.

B. Use the definition for an integral domain to prove that Z7 is an integral domain.

1. State each step of your proof.

2. Provide written justification for each step of your proof.

C. 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.

D. 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.

I look forward to hearing back from you. Thank you!

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