 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: 8th. Contemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708. Since 76 problems in chapter 13 have been answered, more than 15348 students have viewed full stepbystep solutions from this chapter.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

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.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.
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