Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 8.3, Problem 17E
Suppose that
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Chapter 8 Solutions
Elements Of Modern Algebra
Ch. 8.1 - True or False
Label each of the following...Ch. 8.1 - Prob. 2TFECh. 8.1 - Prob. 3TFECh. 8.1 - Prob. 4TFECh. 8.1 - Prob. 5TFECh. 8.1 - Prob. 6TFECh. 8.1 - Prob. 7TFECh. 8.1 - Prob. 1ECh. 8.1 - Prob. 2ECh. 8.1 - Prob. 3E
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 - Prob. 8ECh. 8.1 - Prob. 9ECh. 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 - Prob. 13ECh. 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 - Prob. 17ECh. 8.1 - 18. Let be a commutative ring with unity, and let...Ch. 8.1 - Prob. 19ECh. 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 - Prob. 24ECh. 8.1 - (See exercise 24.) Show that the relation...Ch. 8.2 - Label each of the following statements as either...Ch. 8.2 - Prob. 2TFECh. 8.2 - Prob. 3TFECh. 8.2 - Prob. 1ECh. 8.2 - Prob. 2ECh. 8.2 - Prob. 3ECh. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - Prob. 5ECh. 8.2 - For , , and given in Exercises 1-6, find and in...Ch. 8.2 - Prob. 7ECh. 8.2 - Prob. 8ECh. 8.2 - Prob. 9ECh. 8.2 - Prob. 10ECh. 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 - Prob. 13ECh. 8.2 - Prob. 14ECh. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prob. 17ECh. 8.2 - Prob. 18ECh. 8.2 - Prob. 19ECh. 8.2 - Prob. 20ECh. 8.2 - Prob. 21ECh. 8.2 - Prob. 22ECh. 8.2 - Prob. 23ECh. 8.2 - Prob. 24ECh. 8.2 - Prob. 25ECh. 8.2 - Prob. 26ECh. 8.2 - Prob. 27ECh. 8.2 - Prob. 28ECh. 8.2 - Prob. 29ECh. 8.2 - Prob. 30ECh. 8.2 - Prob. 31ECh. 8.2 - Prob. 32ECh. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Prob. 35ECh. 8.3 - True or False
Label each of the following...Ch. 8.3 - Label each of the following statements as either...Ch. 8.3 - Prob. 3TFECh. 8.3 - True or False
Label each of the following...Ch. 8.3 - Prob. 5TFECh. 8.3 - Prob. 6TFECh. 8.3 - Prob. 7TFECh. 8.3 - True or False
Label each of the following...Ch. 8.3 - Prob. 9TFECh. 8.3 - Prob. 1ECh. 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 - Prob. 10ECh. 8.3 - Prob. 11ECh. 8.3 - Prob. 12ECh. 8.3 - Prob. 13ECh. 8.3 - Prob. 14ECh. 8.3 - Prob. 15ECh. 8.3 - Prob. 16ECh. 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 - Prob. 24ECh. 8.3 - Prob. 25ECh. 8.3 - Prob. 26ECh. 8.3 - Prob. 27ECh. 8.4 - Label each of the following statements as either...Ch. 8.4 - Prob. 2TFECh. 8.4 - Prob. 3TFECh. 8.4 - Prob. 4TFECh. 8.4 - Prob. 5TFECh. 8.4 - Prob. 6TFECh. 8.4 - Prob. 7TFECh. 8.4 - Prob. 8TFECh. 8.4 - Prob. 9TFECh. 8.4 - Prob. 10TFECh. 8.4 - True or False
Label each of the following...Ch. 8.4 - Prob. 12TFECh. 8.4 - Prob. 13TFECh. 8.4 - Prob. 14TFECh. 8.4 - Prob. 15TFECh. 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 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - Prob. 5ECh. 8.4 - Prob. 6ECh. 8.4 - Prob. 7ECh. 8.4 - Prob. 8ECh. 8.4 - Prob. 9ECh. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 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 - Prob. 15ECh. 8.4 - Factors each of the polynomial in Exercise 1316 as...Ch. 8.4 - Prob. 17ECh. 8.4 - Show that the converse of Eisenstein’s...Ch. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Use Theorem to show that each of the following...Ch. 8.4 - Prob. 22ECh. 8.4 - Prove that for complex numbers .
Ch. 8.4 - Prob. 24ECh. 8.4 - Prob. 25ECh. 8.4 - Prob. 26ECh. 8.4 - Prob. 27ECh. 8.4 - Prob. 28ECh. 8.4 - Prob. 29ECh. 8.4 - Prob. 30ECh. 8.4 - Prob. 31ECh. 8.4 - Prob. 32ECh. 8.4 - Let where is a field and let . Prove that if is...Ch. 8.4 - Prob. 34ECh. 8.4 - Prob. 35ECh. 8.5 - Prob. 1TFECh. 8.5 - Prob. 2TFECh. 8.5 - Prob. 3TFECh. 8.5 - Prob. 4TFECh. 8.5 - Prob. 1ECh. 8.5 - Prob. 2ECh. 8.5 - Prob. 3ECh. 8.5 - Prob. 4ECh. 8.5 - Prob. 5ECh. 8.5 - Prob. 6ECh. 8.5 - In Exercises , use the techniques presented in...Ch. 8.5 - Prob. 8ECh. 8.5 - Prob. 9ECh. 8.5 - Prob. 10ECh. 8.5 - Prob. 11ECh. 8.5 - Prob. 12ECh. 8.5 - Prob. 13ECh. 8.5 - Prob. 14ECh. 8.5 - Prob. 15ECh. 8.5 - Prob. 16ECh. 8.5 - Prob. 17ECh. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.5 - Prob. 20ECh. 8.5 - Prob. 21ECh. 8.5 - Prob. 22ECh. 8.5 - Prob. 23ECh. 8.5 - Prob. 24ECh. 8.5 - Prob. 25ECh. 8.5 - Prob. 26ECh. 8.5 - Prob. 27ECh. 8.5 - Prob. 28ECh. 8.5 - Prob. 29ECh. 8.5 - Prob. 30ECh. 8.5 - Derive the quadratic formula by using the change...Ch. 8.5 - Prob. 32ECh. 8.6 - True or False
Label each of the following...Ch. 8.6 - Prob. 2TFECh. 8.6 - Prob. 3TFECh. 8.6 - Prob. 1ECh. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 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 - Prob. 7ECh. 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 - Prob. 10ECh. 8.6 - Prob. 11ECh. 8.6 - Prob. 12ECh. 8.6 - Prob. 13ECh. 8.6 - Prob. 14ECh. 8.6 - Prob. 15ECh. 8.6 - Each of the polynomials in Exercises is...Ch. 8.6 - Prob. 17ECh. 8.6 - Prob. 18E
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- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.arrow_forwardProve Theorem If and are relatively prime polynomials over the field and if in , then in .arrow_forwardUse Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .arrow_forward
- True or False Label each of the following statements as either true or false. Every polynomial equation of degree over a field can be solved over an extension field of .arrow_forwardIf is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .arrow_forwardEach of the polynomials in Exercises is irreducible over the given field . Find all zeros of in the field obtained by adjoining a zero of to . (In Exercises and , has three zeros in .)arrow_forward
- 8. Prove that the characteristic of a field is either 0 or a prime.arrow_forwardLet Q denote the field of rational numbers, R the field of real numbers, and C the field of complex. Determine whether each of the following polynomials is irreducible over each of the indicated fields, and state all the zeroes in each of the fields. a. x22 over Q, R, and C b. x2+1 over Q, R, and C c. x2+x2 over Q, R, and C d. x2+2x+2 over Q, R, and C e. x2+x+2 over Z3, Z5, and Z7 f. x2+2x+2 over Z3, Z5, and Z7 g. x3x2+2x+2 over Z3, Z5, and Z7 h. x4+2x2+1 over Z3, Z5, and Z7arrow_forwardLet ab in a field F. Show that x+a and x+b are relatively prime in F[x].arrow_forward
- Prove that if R is a field, then R has no nontrivial ideals.arrow_forwardProve Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inarrow_forwardLet f(x),g(x) in K[x], a polynomial ring over the field K. Suppose that g(x)=f(ax+b), where a,b ne 0 in K. Prove that f(x) and g(x) have equal splitting fields over K.arrow_forward
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