Reference no: EM13907416

Part 1

Rings are an important algebraic structure, and modular arithmetic has that structure.

Recall that for the mod m relation, the congruence class of an integer xis denoted Nrri, for example, the elements of [-5] are of the form -5 plus integer multiples of 7, which would equate to {... -19, -12, -5, 2, 9, 16, or, more formally, {y: y = -5 + 7g for some integer q).

Task:

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

[a]_i + [b]₇ = [a + b]₇ and [a x b]₇ = [a x b]₇

Note: On the right-hand-side of these equations, + and x 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.

Part 2

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.

C. When you use sources, include all in-text citations and references in APA format.

Part 3

Introduction:

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

Task:

A. let G be the set of die 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 The %Melds of F.
1. Use the definitions of a field and a subfield to prove that S T is a field, showing all work.

C. When you use sources, include all in-text dtations and references in APA format

Note: Please submit all graphics and equations in * .pdf (Portable Document Format) files.

Note: When &dieted points are present th the task prompt, the level of detail or support called for in the rubric refers to those bilikted pods.

Note: For definitions of terms commonly used in the nibric, see the Rubric Terms web link included in the Evaluation Procedures section.

Note: When using sources to support ideas and elements in a paper or project, the submission RUST include APA formatted in-text citations with a corresponding reference kt for any direct quotes or paraphrasing. It is not necessary to list sources that were ccnsutted if they have not been quoted or paraphrased in the text of the paper or Project

Note: No more than a combined total of 30% c If a submission can be directly quoted or closely paraphrased from sources, emen ig cited correctly. For tips on using APA style, please refer to the AM Handout web fink included in the General Instructions section.