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Elements Of Modern Algebra

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

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Chapter
Section
BuyFindarrow_forward

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=[3]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.

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Chapter 8 Solutions

Elements Of Modern Algebra
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Ch. 8.1 - Consider the following polynomial over Z9, where a...Ch. 8.1 - 5. Decide whether each of the following subset is...Ch. 8.1 - Determine which subset in Exercise 5 are ideals of...Ch. 8.1 - Prove that [ x ]={ a0+a1x+...+anxna0=2kfork }, the...Ch. 8.1 - 8. a. Prove that , the set of all polynomials with...Ch. 8.1 - 9. a. Let be a field. Prove that , the set of all...Ch. 8.1 - Let R be a commutative ring with unity. Prove that...Ch. 8.1 - 11. a. List all the polynomials in that have...Ch. 8.1 - a. Find a nonconstant polynomial in Z4[ x ], if...Ch. 8.1 - 13. a. How many polynomials of degree 2 are there...Ch. 8.1 - 14. Prove or disprove that is a field if is a...Ch. 8.1 - 15. Prove that if is an ideal in a commutative...Ch. 8.1 - a. If R is a commutative ring with unity, show...Ch. 8.1 - 17. a. Suppose that is a commutative ring with...Ch. 8.1 - 18. Let be a commutative ring with unity, and let...Ch. 8.1 - In the integral domain Z[ x ], let (Z[ x ])+...Ch. 8.1 - Consider the mapping :Z[ x ]Zk[ x ] defined by...Ch. 8.1 - Describe the kernel of epimorphism in Exercise...Ch. 8.1 - Assume that each of R and S is a commutative ring...Ch. 8.1 - Describe the kernel of epimorphism in Exercise...Ch. 8.1 - For each f(x)=i=0naixi in R[x], the formal...Ch. 8.1 - (See exercise 24.) Show that the relation...Ch. 8.2 - Label each of the following statements as either...Ch. 8.2 - True or False Label each of the following...Ch. 8.2 - True or False Label each of the following...Ch. 8.2 - For,, and given in Exercises , find and in that...Ch. 8.2 - For and given in Exercises , find and in that...Ch. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For , , and given in Exercises 7-10, find the...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For f(x), g(x), and Zn[ x ] given in Exercises...Ch. 8.2 - For , , and given in Exercises , find and in ...Ch. 8.2 - a. Factor x as a product of two polynomials of...Ch. 8.2 - Factor each of the following polynomials as the...Ch. 8.2 - 17. Factor each of the following polynomials as...Ch. 8.2 - Prove or disprove that the polynomial can be...Ch. 8.2 - Let I be the principal ideal (x2+1)={...Ch. 8.2 - 20. Let be the principal ideal . Determine...Ch. 8.2 - 21. Let be the principal ideal . Determine...Ch. 8.2 - 22. Let be the principal ideal . Determine...Ch. 8.2 - 23. Let where . Prove . Ch. 8.2 - Let f(x)=anxn+an1xn1++a0 where an0. Find the...Ch. 8.2 - 25. Prove that if and are nonzero elements of ...Ch. 8.2 - Prove that if d1(x) and d2(x) are monic...Ch. 8.2 - 27. Show that the polynomials and in the...Ch. 8.2 - Prove that if h(x)|f(x) and h(x)|g(x) in F[ x ],...Ch. 8.2 - 29. Let . Prove that if and then . Ch. 8.2 - 30.In the statement of the Division...Ch. 8.2 - 31. With the notation used in the description of...Ch. 8.2 - Prove that every nonzero remainder rj(x) in the...Ch. 8.2 - Prove that the only elements of F[ x ] that have...Ch. 8.2 - Prove that every ideal in where is a field, is a...Ch. 8.2 - Follow the pattern in Definition of Section to...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - Label each of the following statements as either...Ch. 8.3 - Label each of the following statements as either...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - True or False Label each of the following...Ch. 8.3 - 1. Determine the remainder when is divided by ...Ch. 8.3 - Let Q denote the field of rational numbers, R the...Ch. 8.3 - Find all monic irreducible polynomials of degree 2...Ch. 8.3 - Write each of the following polynomials as a...Ch. 8.3 - Let F be a field and f(x)=a0+a1x+...+anxnF[x]....Ch. 8.3 - Prove Corollary 8.18: A polynomial of positive...Ch. 8.3 - Corollary requires that be a field. Show that...Ch. 8.3 - Let be an irreducible polynomial over a field ....Ch. 8.3 - Let be a field. Prove that if is a zero of then...Ch. 8.3 - Let and be two polynomials over the field , both...Ch. 8.3 - Let be a prime integer, and consider the...Ch. 8.3 - Find all the zeros of each of the following...Ch. 8.3 - Give an example of a polynomial of a degree 4 over...Ch. 8.3 - If and are polynomials over the field , and ,...Ch. 8.3 - If f(x) and g(x) are polynomials over the field F,...Ch. 8.3 - Let f(x) be a polynomial of positive degree n over...Ch. 8.3 - Suppose that f(x),g(x), and h(x) are polynomials...Ch. 8.3 - Prove that a polynomial f(x) of positive degree n...Ch. 8.3 - Prove Theorem Suppose is an irreducible...Ch. 8.3 - Prove Theorem If and are relatively prime...Ch. 8.3 - Prove the Unique Factorization Theorem in ...Ch. 8.3 - Let ab in a field F. Show that x+a and x+b are...Ch. 8.3 - Let f(x),g(x),h(x)F[x] where f(x) and g(x) are...Ch. 8.3 - Let where and are relatively prime. If and ,...Ch. 8.3 - Let f(x),g(x),h(x)F[x] where f(x) and g(x) are...Ch. 8.3 - Let and the greatest common divisor of and ...Ch. 8.3 - Find the least common multiple of each pair of...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - Label each of the following statements as either...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - True or False Label each of the following...Ch. 8.4 - 1. Find a monic polynomial of least degree over ...Ch. 8.4 - One of the zeros is given for each of the...Ch. 8.4 - Find all rational zeros of each of the polynomial...Ch. 8.4 - Find all rational zeros of each of the polynomial...Ch. 8.4 - Find all rational zeros of each of the polynomial...Ch. 8.4 - Find all rational zeros of each of the polynomial...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise 712, Find all zeros of the given...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise, Find all zeros of the given...Ch. 8.4 - In Exercise 712, Find all zeros of the given...Ch. 8.4 - Factor each of the polynomial in Exercise as a...Ch. 8.4 - Factor each of the polynomial in Exercise as a...Ch. 8.4 - Factor each of the polynomial in Exercise as a...Ch. 8.4 - Factors each of the polynomial in Exercise 1316 as...Ch. 8.4 - Show that each of the following polynomials is...Ch. 8.4 - Show that the converse of Eisenstein’s...Ch. 8.4 - Let be a polynomial of positive degree with...Ch. 8.4 - Show that each of the following polynomials is...Ch. 8.4 - Use Theorem to show that each of the following...Ch. 8.4 - Show that converse of Theorem is not true by...Ch. 8.4 - Prove that for complex numbers . Ch. 8.4 - Prove that z1z2zn=z1z2zn for complex numbers...Ch. 8.4 - Let be prime. Use Eisenstien’s Criterion to...Ch. 8.4 - Let f(x)=a0+a1x++an1xn1+xn be a monic polynomial...Ch. 8.4 - Derive the quadratic formula for the zeros of...Ch. 8.4 - Prove Theorem . (Hint: In the factorization...Ch. 8.4 - Prove that any polynomial of odd degree that has...Ch. 8.4 - Let in . Prove that if for all or if for all,...Ch. 8.4 - Let in . Prove that if the coefficients ...Ch. 8.4 - Let be in the field. Define the mapping by ....Ch. 8.4 - Let where is a field and let . Prove that if is...Ch. 8.4 - Show that each of the following polynomials is...Ch. 8.4 - Prove that is irreducible over for any prime....Ch. 8.5 - True or False Label each of the following...Ch. 8.5 - True or False Label each of the following...Ch. 8.5 - True or False Label each of the following...Ch. 8.5 - True or False Label each of the following...Ch. 8.5 - In Exercises 118, use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises 118, use the techniques presented in...Ch. 8.5 - In Exercises 118, use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - In Exercises 122, use the techniques presented in...Ch. 8.5 - In exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 1-22, use the techniques presented in...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - In Exercises 23-28, characterize the solutions to...Ch. 8.5 - Prove Theorem 8.38: The change of variable x=ya4...Ch. 8.5 - Show that the change of variable in yields an...Ch. 8.5 - Derive the quadratic formula by using the change...Ch. 8.5 - Use the definition of the discriminant to show...Ch. 8.6 - True or False Label each of the following...Ch. 8.6 - Label each of the following statements as either...Ch. 8.6 - Label each of the following statements as either...Ch. 8.6 - Each of the following polynomials is irreducible...Ch. 8.6 - In each of the following parts, a polynomial p(x)...Ch. 8.6 - In Exercises 36, a field F, a polynomial p(x) over...Ch. 8.6 - In Exercises, a field , a polynomial over , and...Ch. 8.6 - In Exercises , a field , a polynomial over , and...Ch. 8.6 - In Exercises , a field , a polynomial over , and...Ch. 8.6 - For the given irreducible polynomial p(x) over 3,...Ch. 8.6 - If is a finite field with elements, and is a...Ch. 8.6 - Construct a field having the following number of...Ch. 8.6 - Find the multiplicative inverse of in , where is...Ch. 8.6 - Find the multiplicative inverse of 9333+2 in (33),...Ch. 8.6 - An element of a field is a perfect square in if...Ch. 8.6 - Determine whether each of the following...Ch. 8.6 - Find the value of c that will cause the polynomial...Ch. 8.6 - Each of the polynomials in Exercises is...Ch. 8.6 - Each of the polynomials in Exercises is...Ch. 8.6 - Each of the polynomials p(x) in Exercises 1518 is...Ch. 8.6 - Each of the polynomials in Exercises is...

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