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All Textbook Solutions for Elements Of Modern Algebra

19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E Exercise are stated using the notation in the last paragraph of Example. Prove or disprove for all . Example : In this example we outline the development of the quaternions as the set , For each in , we imitate conjugates in and write , It is left as exercise to verify that If , then , and . Thus has all the field properties except commutative multiplication. 42E 43E 44E 45E 46E 47E 48E 49E 50E An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements. Prove that a finite ring R with unity and no zero divisors is a division ring.True or False Label each of the following statements as either true or false. There is a one-to-one correspondence between the standard form and the trigonometric form of a complex number. 2TFE 3TFE 4TFE 1E Find each of the following products. Write each result in both trigonometric and standard form. 3E Show that the n distinct n th roots of 1 are equally spaced around a circle with center at the origin and radius 1.5E 6E 7E 8E 9E 10E 11E 12E 13E 14E Prove that the group in Exercise is cyclic, with as a generator. Prove that for a fixed value of , the set of all th roots of forms a group with respect to multiplication. 16E 17E 18E 19E 20E 21E 22E Prove that the set of all complex numbers that have absolute value forms a group with respect to multiplication. 24E 25E 26E 27E 28E True or False Label each of the following statements as either true or false where represents a commutative ring with unity. A polynomials in over is made up of sums of terms of the form where each and. 2TFE 3TFE 4TFE 5TFE 6TFE 7TFE 1E 2E 3E Consider the following polynomial over Z9, where a is written for [ a ] in Z9: f(x)=2x3+7x+4, g(x)=4x2+4x+6, h(x)=6x2+3. Find each of the following polynomials with all coefficients in Z9. a. f(x)+g(x) b. g(x)+h(x) c. f(x)g(x) d. g(x)h(x) e. f(x)g(x)+h(x) f. f(x)+g(x)h(x) g. f(x)g(x)+f(x)h(x) h. f(x)h(x)+g(x)h(x)5. Decide whether each of the following subset is a subring of , and justify your decision in each case. a. The set of all polynomials with zero constant term. b. The set of all polynomials that have zero coefficients for all even powers of . c. The set of all polynomials that have zero coefficients for all odd powers of . d. The set consisting of the zero polynomials together with all polynomials that have degree 2 or less. Determine which subset in Exercise 5 are ideals of R[x] and which are principal ideals. Justify your choices. Decide whether each of the following subset is a subring of R[x], and justify your decision in each case. The set of all polynomials with zero constant term. The set of all polynomials that have zero coefficients for all even powers of x. The set of all polynomials that have zero coefficients for all odd powers of x. The set consisting of the zero polynomials together with all polynomials that have degree 2 or less.Prove that [ x ]={ a0+a1x+...+anxna0=2kfork }, the set of all polynomials in [ x ] with even constant term, is an ideal of [ x ]. Show that [ x ] is not a principal ideal; that is, show that there is no f(x)[ x ] such that [ x ]=(f(x))={ f(x)g(x)g(x)[ x ] }. Show that [ x ] is an ideal generated by two elements in [ x ] that is, [ x ]=(x,2)={ xf(x)+2g(x)f(x),g(x)[ x ] }.8E 9E Let R be a commutative ring with unity. Prove that deg(f(x)g(x))degf(x)+degg(x) for all nonzero f(x), g(x) in R[ x ], even if R in not an integral domain.11. a. List all the polynomials in that have degree 2. b. Determine which of the polynomials in part a are units. If none exists, state so. a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant polynomial in Z3[ x ], if one exists, that is a unit. c. Prove or disprove that there exist nonconstant polynomials in Zp[ x ] that are units if p is prime.13E 14. Prove or disprove that is a field if is a field. 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in . a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].17E 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to . 19E Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.Describe the kernel of epimorphism in Exercise 22. Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn.24E (See exercise 24.) Show that the relation f(x)Rg(x) if and only if f(x)=g(x) is an equivalence relation on R[ x ]. Describe the equivalence class [ f(x) ]. For each f(x)=i=0naixi in R[ x ], the formal derivative of f(x) is the polynomial f(x)=i=1niaixi1. (For n=0, f(x)=0 by definition.) Prove that [ f(x)+g(x) ]=f(x)+g(x). Prove that [ f(x)g(x) ]=f(x)g(x)+f(x)g(x).Label each of the following statements as either true or false. Every f(x) in F(x), where F is a field, can be factored.2TFE 3TFE 1E 2E 3E For , , and given in Exercises 1-6, find and in that satisfy the conditions in the Division Algorithm. 4. , , in 5E For , , and given in Exercises 1-6, find and in that satisfy the conditions in the Division Algorithm. 6. , , in 7E 8E 9E 10E For f(x), g(x), and Zn[ x ] given in Exercises 11-14, find s(x) and t(x) in Zn[ x ] such that d(x)=f(x)s(x)+g(x)t(x) is the greatest common divisor of f(x) and g(x). f(x)=2x3+2x2+x+1, g(x)=x4+2x2+x+1, in Z3[ x ]For f(x), g(x), and Zn[ x ] given in Exercises 11-14, find s(x) and t(x) in Zn[ x ] such that d(x)=f(x)s(x)+g(x)t(x) is the greatest common divisor of f(x) and g(x). f(x)=2x3+x2+1, g(x)=x5+x4+2x2+1, in Z3[ x ]13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E True or False Label each of the following statements as either true or false. For each in a field , the value is unique, where Label each of the following statements as either true or false. We say that cF is a solution to the polynomial equation f(x)=0 if and only if f(c)=0inF.3TFE True or False Label each of the following statements as either true or false. 4. Any polynomial of positive degree over the field has exactly distinct zeros in . 5TFE 6TFE 7TFE True or False Label each of the following statements as either true or false. 8. Any polynomial of positive degree that is reducible over a field has at least one zero in . 9TFE 1E Let 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 Z7 Find all monic irreducible polynomials of degree 2 over Z3.Write each of the following polynomials as a products of its leading coefficient and a finite number of monic irreducible polynomials over 5. State their zeros and the multiplicity of each zero. 2x3+1 3x3+2x2+x+2 3x3+x2+2x+4 2x3+4x2+3x+1 2x4+x3+3x+2 3x4+3x3+x+3 x4+x3+x2+2x+3 x4+x3+2x2+3x+2 x4+2x3+3x+4 x5+x4+3x3+2x2+4x Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in Corollary requires that be a field. Show that each of the following polynomials of positive degree has more than zeros over where is not a field. over over Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in Let be a field. Prove that if is a zero of then is a zero of 10E 11E 12E 13E 14E 15E 16E Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some . Prove Theorem If and are relatively prime polynomials over the field and if in , then in . Prove the Unique Factorization Theorem in (Theorem). Theorem Unique Factorisation Theorem Every polynomial of positive degree over the field can be expressed as a product of its leading coefficient and a finite number of monic irreducible polynomials over . This factorization is unique except for the order of the factors. Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].Let f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.24E 25E 26E 27E Label each of the following statements as either true or false. 1. Every polynomial of positive degree over the complex numbers has a zero in the complex numbers. 2TFE 3TFE 4TFE 5TFE 6TFE 7TFE 8TFE 9TFE 10TFE True or False Label each of the following statement as either true or false. 11. Every primitive polynomial is monic. 12TFE 13TFE 14TFE 15TFE 1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic polynomial of least degree with real coefficients that has the given numbers as zeros. a. b. c. d. e. f. g. and h. and One of the zeros is given for each of the following polynomial. Find the other zeros in the field of complex numbers. is a zero. is a zero. is a zero is a zero. 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E Factor each of the polynomial in Exercise as a product of its leading coefficient and a finite number of monic irreducible polynomial over the field of rational numbers. Factor each of the polynomial in Exercise as a product of its leading coefficient and a finite number of monic irreducible polynomial over the field of rational numbers. 15E Factors each of the polynomial in Exercise 1316 as a product of its leading coefficient and a finite number of monic irreducible polynomial over the field of rational numbers. 6x4+x3+3x214x8 17E Show that the converse of Eisenstein’s Irreducibility Criterion is not true by finding an irreducible such that there is no that satisfies the hypothesis of Eisenstein’s Irreducibility Criterion. 19E 20E Use 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 . 22E Prove that for complex numbers . 24E 25E 26E 27E 28E 29E 30E 31E 32E Let where is a field and let . Prove that if is irreducible over , then is irreducible over . 34E 35E 1TFE 2TFE 3TFE 4TFE 1E 2E 3E 4E 5E 6E In Exercises , use the techniques presented in this section to find all solutions of the given equation. 7. 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E Derive the quadratic formula by using the change in variable x=y12(ba) to transform the quadratic equation x2+bax+ca=0 into one involving the difference of two squares and solve the resulting equation.32E 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 . 2TFE 3TFE 1E 2E 3E In Exercises, a field , a polynomial over , and an element of the field obtained by adjoining a zero of to are given. In each case: Verify that is irreducible over . Write out a formula for the product of two arbitrary elements and of . Find the multiplicative inverse of the given element of . , , In Exercises , a field , a polynomial over , and an element of the field obtained by adjoining a zero of to are given. In each case: Verify that is irreducible over . Write out a formula for the product of two arbitrary elements and of . Find the multiplicative inverse of the given element of . , , In Exercises , a field , a polynomial over , and an element of the field obtained by adjoining a zero of to are given. In each case: Verify that is irreducible over . Write out a formula for the product of two arbitrary elements and of . Find the multiplicative inverse of the given element of . , , 7E If 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 . Construct a field having the following number of elements. 10E 11E 12E 13E 14E 15E Each 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 .) 17E 18E

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