 18.1CE: Exercise 1. In the ring Z[i] (where i2 = 1) this software determin...
 18.1E: For the ring , where d ? 1 and d is not divisible by the square of ...
 18.1SE: Suppose that F is a field and there is a ring homomorphism from Z o...
 18.2E: In an integral domain, show that a and b are associates if and only if
 18.2SE: Let Determine all ring automorphisms of
 18.3E: Show that the union of a chain I1 ? I2 ? · · · of ideals of a ring ...
 18.3SE: (Second Isomorphism Theorem for Rings) Let A be a subring of R and ...
 18.4E: In an integral domain, show that the product of an irreducible and ...
 18.4SE: (Third Isomorphism Theorem for Rings) Let A and B be ideals of a ri...
 18.5E: Suppose that a and b belong to an integral domain, b ? 0, and a is ...
 18.5SE: Let f(x) and g(x) be irreducible polynomials over a field F. If f(x...
 18.6E: Let D be an integral domain. Define a?b if a and b are associates. ...
 18.6SE: (Chinese Remainder Theorem for Rings) If R is a commutative ring an...
 18.7E: In the notation of Example 7, show that d(xy) = d(x)d(y).Reference:
 18.7SE: Prove that the set of all polynomials whose coefficients are all ev...
 18.8E: Let D be a Euclidean domain with measure d. Prove that u is a unit ...
 18.8SE: Let Show that I is a maximal ideal of R.
 18.9E: Let D be a Euclidean domain with measure d. Show that if a and b ar...
 18.9SE: Let R be a ring with unity and let a be a unit in R. Show that the ...
 18.10E: Let D be a principal ideal domain and let p ? D. Prove that is a ma...
 18.10SE: Let with b ? 0. Show that 2 does not belong to
 18.11E: Trace through the argument given in Example 7 to find q and r in Z[...
 18.11SE: Show that is a field. How many elements does it have?
 18.12E: Let D be a principal ideal domain. Show that every proper ideal of ...
 18.12SE: Is the homomorphic image of a principal ideal domain a principal id...
 18.13E: In show that 21 does not factor uniquely as a product of irreducibles.
 18.13SE: For any f(x) ? Zp[x], show that f(xp) = ( f(x)) p.
 18.14E: Show that 1 –i is an irreducible in Z[i].
 18.14SE: Let p be a prime. Show that there is exactly one ring homomorphism ...
 18.15E: Show that is not a unique factorization domain. (Hint: Factor 10 in...
 18.15SE: Recall that a is an idempotent if a2 = a. Show that if 1 + k is an ...
 18.16E: Give an example of a unique factorization domain with a subdomain t...
 18.16SE: Show that Zn (where n> 1) always has an even number of idempotents....
 18.17E: In Z[i], show that 3 is irreducible but 2 and 5 are not.
 18.17SE: Show that the equation x2 + y2 = 2003 has no solutions in the integ...
 18.18E: Prove that 7 is irreducible in even though N(7) is not prime.
 18.18SE: Prove that if both k and k + 1 are idempotents in Zn and k ? 0, the...
 18.19E: Prove that if p is a prime in Z that can be written in the form a2 ...
 18.19SE: Prove that x4 + 15x3 + 7 is irreducible over Q.
 18.20E: Prove that is not a principal ideal domain.
 18.20SE: For any integers m and n, prove that the polynomial x3 + (5m + 1)x ...
 18.21E: In is irreducible but not prime.
 18.21SE: Prove that . How many elements are in the ring
 18.22E: In are irreducible but not prime.
 18.22SE: Prove that are unique factorization domains. (Hint: Mimic Example 7...
 18.23E: Prove that is not a unique factorization domain.
 18.23SE: Is a maximal ideal in Z[i]?
 18.24E: Let F be a field. Show that in F[x] a prime ideal is a maximal ideal.
 18.24SE: Express both 13 and 5 + i as products of irreducibles from Z[i].
 18.25E: Let d be an integer less than –1 that is not divisible by the squar...
 18.25SE: Let R = {a/b  a, b ? Z, 3 ? b}. Prove that R is an integral domain...
 18.26E: In show that every element of the form is a unit, where n is a posi...
 18.26SE: Give an example of a ring that contains a subring isomorphic to Z a...
 18.27E: If a and b belong to where d is not divisible by the square of a pr...
 18.27SE: Show that is not ringisomorphic to Z3 ? Z3.
 18.28E: For a commutative ring with unity we may define associates, irreduc...
 18.28SE: For any n > 1, prove that R is ring isomorphic to Zn ? Zn.
 18.29E: Let n be a positive integer and p a prime that divides n. Prove tha...
 18.29SE: Suppose that R is a commutative ring and I is an ideal of R. Prove ...
 18.30E: Let p be a prime divisor of a positive integer n. Prove that p is i...
 18.30SE: Find an ideal I of Z8[x] such that the factor ring Z8[x]/I is a field.
 18.31E: Prove or disprove that if D is a principal ideal domain, then D[x] ...
 18.31SE: Find an ideal I of Z8[x] such that the factor ring Z8[x]/I is an in...
 18.32E: Determine the units in Z[i].
 18.32SE: Find an ideal I of Z[x] such that Z[x]/I is ringisomorphic to Z3
 18.33E: Let p be a prime in an integral domain. If p  a1a2 · · · an, prove...
 18.34E: Show that 3x2 + 4x + 3 ? Z5[x] factors as (3x + 2)(x + 4) and (4x +...
 18.35E: Let D be a principal ideal domain and p an irreducible element of D...
 18.36E: Show that an integral domain with the property that every strictly ...
 18.37E: An ideal A of a commutative ring R with unity is said to be finitel...
 18.38E: Prove or disprove that a subdomain of a Euclidean domain is a Eucli...
 18.39E: Show that for any nontrivial ideal I of Z[i], Z[i]/I is finite.
 18.40E: Find the inverse of What is the multiplicative order of
 18.41E: In are not associates.
 18.42E: Let R = Z ? Z ? · · · (the collection of all sequences of integers ...
 18.43E: Prove that in a unique factorization domain, an element is irreduci...
 18.44E: Let F be a field and let R be the integral domain in F [x] generate...
 18.45E: Prove that for every field F, there are infinitely many irreducible...
 18.46E: Find a mistake in the statement shown in Figure 18.2.
Solutions for Chapter 18: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 18
Get Full SolutionsThis textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8th. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 18 includes 79 full stepbystep solutions. Contemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708. Since 79 problems in chapter 18 have been answered, more than 17656 students have viewed full stepbystep solutions from this chapter.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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