 13.1CE: Exercise 1. This software lists all idempotents (see the chapter ex...
 13.1E: Verify that Examples 1 through 8 are as claimed.Reference:EXAMPLE 1...
 13.2CE: Exercise 1. This software lists all idempotents (see the chapter ex...
 13.2E: Which of Examples 1 through 5 are fields?Reference:EXAMPLE 1 The ri...
 13.3CE: Exercise 1. This software lists all idempotents (see the chapter ex...
 13.3E: Show that a commutative ring with the cancellation property (under ...
 13.4CE: This software finds the idempotents in Zn[i] = {a+bi a, b belong t...
 13.4E: List all zerodivisors in Z20. Can you see a relationship between t...
 13.5CE: This software finds the nilpotent elements in Z n [ i ] = { a + bi ...
 13.5E: Show that every nonzero element of Zn is a unit or a zerodivisor.
 13.6CE: This software determines the zerodivisors in Zn [i] = { a+bi  a, ...
 13.6E: Find a nonzero element in a ring that is neither a zerodivisor nor...
 13.7E: Let R be a finite commutative ring with unity. Prove that every non...
 13.8E: Let a ? 0 belong to a commutative ring. Prove that a is a zero divi...
 13.9E: Find elements a, b, and c in the ring Z ? Z ? Z such that ab, ac, a...
 13.10E: Describe all zerodivisors and units of Z ? Q ? Z.
 13.11E: Let d be an integer. Prove that is an integral domain. (This exerci...
 13.12E: In Z7, give a reasonable interpretation for the expressions 1/2, .
 13.13E: Give an example of a commutative ring without zerodivisors that is...
 13.14E: Find two elements a and b in a ring such that both a and b are zero...
 13.15E: Let a belong to a ring R with unity and suppose that an = 0 for som...
 13.16E: Show that the nilpotent elements of a commutative ring form a subring.
 13.17E: Show that 0 is the only nilpotent element in an integral domain.
 13.18E: A ring element a is called an idempotent if a2 = a. Prove that the ...
 13.19E: Let a and b be idempotents in a commutative ring. Show that each of...
 13.20E: Show that Zn has a nonzero nilpotent element if and only if n is di...
 13.21E: Let R be the ring of realvalued continuous functions on [–1, 1]. S...
 13.22E: Prove that if a is a ring idempotent, then an – a for all positive ...
 13.23E: Determine all ring elements that are both nilpotent elements and id...
 13.24E: Find a zerodivisor in Z5[i] = {a + bi  a, b ? Z5}.
 13.25E: Find an idempotent in Z5[i] = {a + bi  a, b ? Z5}.
 13.26E: Find all units, zerodivisors, idempotents, and nilpotent elements ...
 13.27E: Determine all elements of a ring that are both units and idempotents.
 13.28E: Let R be the set of all realvalued functions defined for all real ...
 13.29E: (Subfield Test) Let F be a field and let K be a subset of F with at...
 13.30E: Let d be a positive integer. Prove that a, b ? Q} is a field.
 13.31E: Let R be a ring with unity 1. If the product of any pair of nonzero...
 13.32E: Let R = {0, 2, 4, 6, 8} under addition and multiplication modulo 10...
 13.33E: Formulate the appropriate definition of a subdomain (that is, a “su...
 13.34E: Prove that there is no integral domain with exactly six elements. C...
 13.35E: Let F be a field of order 2n. Prove that char F = 2.
 13.36E: Determine all elements of an integral domain that are their own inv...
 13.37E: Characterize those integral domains for which 1 is the only element...
 13.38E: Determine all integers n > 1 for which (n – 1)! is a zerodivisor i...
 13.39E: Suppose that a and b belong to an integral domain.a. If a5 = b5 and...
 13.40E: Find an example of an integral domain and distinct positive integer...
 13.41E: If a is an idempotent in a commutative ring, show that 1 – a is als...
 13.42E: Construct a multiplication table for Z2[i], the ring of Gaussian in...
 13.43E: The nonzero elements of Z3[i] form an Abelian group of order 8 unde...
 13.44E: Show that is a field. For any positive integer k and any prime p, d...
 13.45E: Show that a finite commutative ring with no zerodivisors and at le...
 13.46E: Suppose that a and b belong to a commutative ring and ab is a zero...
 13.47E: Suppose that R is a commutative ring without zerodivisors. Show th...
 13.48E: Suppose that R is a commutative ring without zerodivisors. Show th...
 13.49E: Let x and y belong to a commutative ring R with prime characteristi...
 13.50E: Let R be a commutative ring with unity 1 and prime characteristic. ...
 13.51E: Show that any finite field has order pn, where p is a prime. Hint: ...
 13.52E: Give an example of an infinite integral domain that has characteris...
 13.53E: Let R be a ring and let M2(R) be the ring of 2 × 2 matrices with en...
 13.54E: Let R be a ring with m elements. Show that the characteristic of R ...
 13.55E: Explain why a finite ring must have a nonzero characteristic.
 13.56E: Find all solutions of x2 – x + 2 = 0 over Z3[i]. (See Example 9.)
 13.57E: Consider the equation x2 – 5x + 6 = 0.a. How many solutions does th...
 13.58E: Find the characteristic of Z4 ? 4Z.
 13.59E: Suppose that R is an integral domain in which 20 · 1 = 0 and 12 · 1...
 13.60E: In a commutative ring of characteristic 2, prove that the idempoten...
 13.61E: Describe the smallest subfield of the field of real numbers that co...
 13.62E: Let F be a finite field with n elements. Prove that xn–1 = 1 for al...
 13.63E: Let F be a field of prime characteristic p. Prove that K = {x ?F  ...
 13.64E: Suppose that a and b belong to a field of order 8 and that a2 + ab ...
 13.65E: Let F be a field of characteristic 2 with more than two elements. S...
 13.66E: Suppose that F is a field with characteristic not 2, and that the n...
 13.67E: Suppose that D is an integral domain and that ? is a nonconstant fu...
 13.68E: Let F be a field of order 32. Show that the only subfields of F are...
 13.69E: Suppose that F is a field with 27 elements. Show that for every ele...
 13.70E: Let with the usual matrix addition and multiplication and mod 7 add...
Solutions for Chapter 13: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 13
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 13 includes 76 full stepbystep solutions. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708. Since 76 problems in chapter 13 have been answered, more than 45675 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.