 # Rings and Fields

## Question

Homework #2

Introduction:

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 x is denoted [x]m. For example, the elements of [–5]7 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 + 7q for some integer q}.

A. Use the definition for a ring to prove that Z7 is a ring under the operations + and × defined as follows:
[a]7 + [b]7 = [a + b]7 and [a]7 × [b]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.

Homework #3

Introduction:

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.

## Solution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden. \$13.00 for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

### Find A Tutor

View available Abstract Algebra Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.