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mathematics
a first course in abstract algebra

A First Course In Abstract Algebra 7th Edition John Fraleigh - Solutions

Let R be a commutative ring and N an ideal of R. Referring to Exercise 30, show that if every element of N is nilpotent and the nilradical of R/N is R/N, then the nilradical of R is R.Data from Exercise 30An element a of a ring R is nilpotent if an= 0 for some n ∈ Z+. Show that the collection of
Referring to Exercise 32, show that S ⊆ V(I(S)).Data from Exercise 32Let F be a field. Show that if S is a nonempty subset of Fn, then I(S) = {f(x) ∈ F[x]|f(s) = 0 for all s ∈ S} is an ideal of F[x].
There is a sort of arithmetic of ideals in a ring. The exercises define sum, product, and quotient of ideals. If A and B are ideals of a ring R, the sum A + B of A and B is defined by A+ B = {a+ b | a ∈ A, b ∈ B}.a. Show that A + B is an ideal. b. Show that A ⊆ A + B and B ⊆ A + B.
Let R be a commutative ring and N an ideal of R. Show that the set √N of all a ∈ R, such that an ∈ N for some n ∈ Z+, is an ideal of R, the radical of N.
Referring to Exercise 32, give an example of a subset S of R2 such that V(I(S)) ≠ S.Data from Exercise 32Let F be a field. Show that if S is a nonempty subset of Fn, then I(S) = {f(x) ∈ F[x]|f(s) = 0 for all s ∈ S} is an ideal of F[x].
Show that for a field F, the set S of all matrices of the form for a, b ∈ F is a right ideal but not a left ideal of M2(F). That is, show that S is a subring closed under multiplication on the right by any element of M2(F), but is not closed under left multiplication. b (86) 0 0
Referring to Exercise 34, show by examples that for proper ideals N of a commutative ring R, a. √N need not equal N b. √N may equal N.Data from Exercise 34Let R be a commutative ring and N an ideal of R. Show that the set √N of all a ∈ R, such that an ∈ N for some n ∈ Z+, is
Show that ∅ : C → M2(R) given byfor a, b ∈ R gives an isomorphism of C with the subring ∅[C] of M2(R). p(a + bi) = D -b a
Referring to Exercise 32, show that if N is an ideal of F[x], then N ⊆ I(V(N)).Data from Exercise 32Let F be a field. Show that if S is a nonempty subset of Fn, then I(S) = {f(x) ∈ F[x]|f(s) = 0 for all s ∈ S} is an ideal of F[x].
There is a sort of arithmetic of ideals in a ring. The exercises define sum, product, and quotient of ideals.Let A and B be ideals of a commutative ring R. The quotient A : B of A by B is defined by A : B = {r ∈ R| rb ∈ A for all b ∈ B}. Show that A : B is an ideal of R.
What is the relationship of the ideal √N of Exercise 34 to the nilradical of R/N (see Exercise 30)? Word your answer carefully.Data from Exercise 34Let R be a commutative ring and N an ideal of R. Show that the set √N of all a ∈ R, such that an ∈ N for some n ∈ Z+, is an ideal of R, the
Referring to Exercise 32, give an example of an ideal N in R[x, y] such that I(V(N)) ≠ N.Data from Exercise 32Let F be a field. Show that if S is a nonempty subset of Fn, then I(S) = {f(x) ∈ F[x]|f(s) = 0 for all s ∈ S} is an ideal of F[x].
Show that the matrix ring M2(Z2) is a simple ring; that is, M2(Z2) has no proper nontrivial ideals.
Let R be a ring with unity and let End((R, +)) be the ring of endomorphisms of (R, +). Let a ∈ R, and let λa : R → R be given by λa(x) = ax for x ∈ R. a. Show that λa is an endomorphism of (R, +). b. Show that R' = {λa | a ∈ R} is a subring of End ((R, +)). c. Prove the
Prove that if F is a field, every proper nontrivial prime ideal of F[x] is maximal.
Find q(x) and r(x) as described by the division algorithm so that f (x) = g(x)q(x) + r(x) with r(x) = 0 or of degree less than the degree of g(x).f(x) = x4 + 5x3 - 3x2 and g(x) = 5x2 - x + 2 in Z11[x].
Find the characteristic of the given ring.Z3 x Z3
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field.Z x Z
Compute φ(pq) where both p and q are primes.
Find the center of the group (H*, •), where H* is the set of nonzero quaternions.
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field.{a + b
Describe all solutions of the given congruence.22x ≡ 5 (mod 15)
Show each of the following by giving an example. a. A polynomial of degree n with coefficients in a strictly skew field may have more than n zeros in the skew field. b. A finite multiplicative subgroup of a strictly skew field need not be cyclic.
Let R be a commutative ring with unity of characteristic 3. Compute and simplify (a + b )6 for a, b ∈ R.
Is 2x3 + x2 + 2x + 2 an irreducible polynomial in Z5[x]? Why? Express it as a product of irreducible polynomials in Z5[x].
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field.The set
Find all zeros in the indicated finite field of the given polynomial with coefficients in that field.x3 + 2x + 2 in Z7
With reference to Exercise 12, how many elements are there in the ring Q(Z4 , {1, 3})?Data from Exercise 12Let R be a nonzero commutative ring, and let T be a nonempty subset of R closed under multiplication and containing neither 0 nor divisors of 0. Starting with R x T and otherwise exactly
Redraw Fig. 19.10 to include a subset corresponding to strictly skew fields.Data from Figure 19.10 Commutative rings 4 3 Integral Fields 1 Domains 2 5 Rings with unity 6
Let F be a field and let f (x) ∈ F[x]. A zero of f (x) is an α ∈ F such that ∅α(x)) = 0, where ∅α : F(x) → F is the evaluation homomorphism mapping x into α.
Give a one-sentence synopsis of the proof of Theorem 19 .11. Data from Theorem 19.11 Every finite integral domain is a field. Proof Let 0, 1, a1, ........anbe all the elements of a finite domain D. We need to show that for a ∈ D, where a ≠ 0, there exists b ∈ D such that
Prove that if D is an integral domain, then D [ x] is an integral domain.
Give a synopsis of the proof of Corollary 23.5.Data from 23.5 Corollary A nonzero polynomial f(x) ∈ F[x] of degree n can have at most n zeros in a field F. Proof The preceding corollary shows that if a1 ∈ F is a zero off (x ), then f(x) = (x - a1)q1(x), where, of course, the degree
Show that the multiplication defined on the set F of functions in Example 18.4 satisfies axioms R2 and R3 for a ring.Data from 18.4 Example Let F be the set of all functions f: R → R We know that (F, +) is an abelian group under the usual function addition, (f + g)(x) = f(x) +
An element a of a ring R is idempotent if a2 = a. a. Show that the set of all idempotent elements of a commutative ring is closed under multiplication. b. Find all idempotents in the ring Z6 x Z12.
Partition the following collection of groups into subcollections of isomorphic groups. Here a * superscript means all nonzero elements of the set.Z under addition S2 Z6
Find both the center Z(D4) and the commutator subgroup C of the group D4 of symmetries of the square in Table 8.12.Data from Table 8.12 Po PI P2 P3 f| f2 8₁ 82 Po Po P₁ P2 fy 12. 8₁ P3 Po PI 8₂ P2 P₁ P2 P3 82 P2 P3 Po 8₁ fi P3 fl2 8₁ fly 8₂ M2 8₁ P3 Po Pi fl2
Determine whether the given map ∅ is a homomorphism.Let ∅ : R → Z under addition be given by ∅(x) = the greatest integer ≤ x.
Find the number of distinguishable ways the edges of a square of cardboard can be painted if six colors of paint are available and a. No color is used more than once. b. The same color can be used on any number of edges.
To what group mentioned in the text is the additive group R/Z isomorphic?
Let ∅ : G →G' be a group homomorphism. Show that if |G| is finite, then |∅ [ G] I is finite and is a divisor of |G|.
Let ∅ : G →G' be a group homomorphism. Show that if |G'| is finite, then |∅ [ G] I is finite and is a divisor of |G'|.
Describe the field F of quotients of the integral subdomain D = {n + mi |n, m ∈ Z} of C. "Describe" means give the elements of CC that make up the field of quotients of D in C. (The elements of D are the Gaussian integers.)
Compute the product in the given ring.(12)(16) in Z24
Computations Find all solutions of the equation x3 - 2x2 - 3x = 0 in Z12.
Find the sum and the product of the given polynomials in the given polynomial ring. f(x) = 4x - 5, g(x) = 2x2 - 4x + 2 in Z8[x].
Computations We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a generator for this group for the given finite field.Z7
let G = {e, a, b} be a cyclic group of order 3 with identity element e. Write the element in the group algebra Z5G in the form re + sa + tb for r, s, t ∈ Z5. (2e + 3a + 0b)+(4e + 2a + 3b)
Let F be an ordered field and let F ( (x)) be the field of formal Laurent series with coefficients in F, discussed in Example 25.9. Describe the ordering of the monomials.• -x-3, x-2, x-1, x0 = 1, x, x2, x3, •••in the ordering of F((x)) described in that example.Data from Example
Find q(x) and r(x) as described by the division algorithm so that f (x) = g(x)q(x) + r(x) with r(x) = 0 or of degree less than the degree of g(x).f (x) = x6 + 3x5 + 4x2 - 3x + 2 and g(x) = x2 + 2x - 3 in Z7 [x].
Describe (in the sense of Exercise 1) the field F of quotients of the integral subdomain D = {n + m√2| n, m ∈ Z} of R.Data from Exercise 1Describe the field F of quotients of the integral subdomain D = {n + mi |n, m ∈ Z} of C. "Describe" means give the elements of CC that make up
ComputationsSolve the equation 3x = 2 in the field Z7; in the field Z23.
Compute the product in the given ring.(16)(3) in Z32
Find the sum and the product of the given polynomials in the given polynomial ring. f(x) = x + 1, g(x) = x + 1 in Z2[x].
Computations We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a generator for this group for the given finite field.Z11
let G = {e, a, b} be a cyclic group of order 3 with identity element e. Write the element in the group algebra Z5G in the form re + sa + tb for r, s, t ∈ Z5.(2e + 3a + 0b)(4e + 2a + 3b)
Find q(x) and r(x) as described by the division algorithm so that f (x) = g(x)q(x) + r(x) with r(x) = 0 or of degree less than the degree of g(x).f(x) = x6 + 3x5 +,4x2 - 3x + 2 and g(x) = 3x2 + 2x - 3 in Z7[x].
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. A field of quotients of an integral domain D is a field F in which D can be embedded so that every nonzero element of D is a unit in F.
ComputationsFind all solutions of the equation x2 + 2x + 2 = 0 in Z6.
Compute the product in the given ring.(11)(-4)inZ15
Find the sum and the product of the given polynomials in the given polynomial ring. f(x) = 2x2 + 3x + 4, g(x) = 3x2 + 2x + 3 in Z6 [x].
Example 25.12 described an ordering of Z[√2] = {m + n√2 |m, n ∈ Z} in which -√2 is positive. Describe, in terms of m and n, all positive elements of Z[√2] in that ordering.Data from Example 25.12Exercise 11 of Section 18 shows that {m + n√2| m, n ∈ Z} is a ring. Let us denote this
Computations We will see later that the multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a generator for this group for the given finite field.Z17
Find q(x) and r(x) as described by the division algorithm so that f (x) = g(x)q(x) + r(x) with r(x) = 0 or of degree less than the degree of g(x).f(x) = x5 - 2x4 +3x - 5 and g(x) = 2x + 1 in Z11[x].
Mark each of the following true or false. ___ a. Q is a field of quotients of Z. ___ b. R is a field of quotients of Z. ___ c. R is a field of quotients of R ___ d. C is a field of quotients of R. ___ e. If D is a field, then any field of quotients of D is isomorphic to
ComputationsFind all solutions of x2 + 2x + 4 = 0 in Z6.
Compute the product in the given ring.(20)(-8)inZ26
Find the sum and the product of the given polynomials in the given polynomial ring. f(x) = 2x3 + 4x2 + 3x + 2, g(x) = 3x4 + 2x + 4 in Z5[x].
Using Fermat's theorem, find the remainder of 347 when it is divided by 23.
Write the element of H in the form a1 + a2i + a3j + a4k for ai ∈ R.(i + 3j)(4 + 2j - k)
Find all generators of the cyclic multiplicative group of units of the given finite field. (Review Corollary 6.16.) Z5Data from 6.16 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form ar, where r is
Show by an example that a field F' of quotients of a proper subdomain D' of an integral domain D may also be a field of quotients for D.
Compute the product in the given ring.(2,3)(3,5) in Z5 x Z9
Find the characteristic of the given ring.2Z
How many polynomials are there of degree ≤ 3 in Z2[x]? (Include 0.)
Use Fermat's theorem to find the remainder of 3749 when it is divided by 7.
Write the element of H in the form a1 + a2i + a3j + a4k for ai ∈ R.i2j3kji5
Find all generators of the cyclic multiplicative group of units of the given finite field. (Review Corollary 6.16.) Z7Data from 6.16 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form ar, where r is relatively
Compute the product in the given ring.(-3,5)(2,-4) in Z4 X Z11
Find the characteristic of the given ring.Z x Z
How many polynomials are there of degree ≤ 2 in Z5[x]? (Include 0.)
Compute the remainder of 2(217) + 1 when divided by 19.
Write the element of H in the form a1 + a2i + a3j + a4k for ai ∈ R.(i + j)-1
Find all generators of the cyclic multiplicative group of units of the given finite field. (Review Corollary 6.16.) Z17Data from 6.16 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form ar, where r is relatively
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field.nZ with
Find the characteristic of the given ring.Z3 x 3Z
Make a table of values of φ(n) for n ≤ 30.
Write the element of H in the form a1 + a2i + a3j + a4k for ai ∈ R.[(1 + 3i)(4j + 3k)]-1
Find all generators of the cyclic multiplicative group of units of the given finite field. (Review Corollary 6.16.) Z23Data from 6.16 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form ar, where r is relatively
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field.Z+ with
Compute φ(p2) where p is a prime.
The polynomial x4 + 4 can be factored into linear factors in Z5[x]. Find this factorization.
Find the characteristic of the given ring.Z3 x Z4
Prove Part 6 of Step 3. You may assume any preceding part of Step 3.
The polynomial x3 + 2x2 + 2x + 1 can be factored into linear factors in Z7 [x]. Find this factorization.
Find the characteristic of the given ring.Z6 x Z15
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field.2Z x Z
Use Euler's generalization of Fermat's theorem to find the remainder of 71000 when divided by 24.
Find two subsets of H different from C and from each other, each of which is a field isomorphic to C under the induced addition and multiplication from H.
Let R be a commutative ring with unity of characteristic 4. Compute and simplify (a + b )4 for a, b ∈ R.

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