   # Construct a field having the following number of elements. 2 4 5 2 3 3 7 2 ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

#### Solutions

Chapter
Section ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 8.6, Problem 9E
Textbook Problem
1 views

## Construct a field having the following number of elements. 2 4 5 2 3 3 7 2

(a)

To determine

A field having 24 number of elements.

### Explanation of Solution

Given information:

n=24

Formula used:

(i) Let p(x) be a polynomial of positive degree over the field F. Then the ring F[x]/(p(x)) is a field if and only if p(x) is an irreducible polynomial over F.

(ii) Elements of F[x]/(p(x)):

Let p(x) be a polynomial of positive degree n over the field F, let P=(p(x)), and let α=x+P in F(x)/P. Then each element of the ring F(x)/P can be expressed uniquely in the form a0+a1α+a2α2+...+an1αn1

Explanation:

Let p(x)=x4+x+12[x]

2={0,1}

p(0)=04+0+1=10

p(1)=14+1+1=2=10

Hence, the polynomial p(x)=x4+x+1 is irreducible over 2

Thus 2[x]p(x) is a field.

Let x=α be a zero of p(x) in the field 2[x]/p(x)

(b)

To determine

A field having 52 number of elements.

(c)

To determine

A field having 33 number of elements.

(d)

To determine

A field having 72 number of elements.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts
In Problems 45-51, solve each system of equations. 51.

Mathematical Applications for the Management, Life, and Social Sciences

Evaluate the indefinite integral. cos(/x)x2dx

Single Variable Calculus: Early Transcendentals, Volume I

Evaluate the expression sin Exercises 116. (2)3

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 39-44, write the equation in the slope-intercept form and then find the slope and y-intercept of t...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

True or False: If a number p is in the interval of convergence of then so is the number .

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Show that lnex=x

College Algebra (MindTap Course List) 