 17.1CE: Exercise 1. This software implements the "Mod p irreducibility Test...
 17.1E: Verify the assertion made in Example 2.
 17.2E: Suppose that D is an integral domain and F is a field containing D....
 17.3E: Show that a nonconstant polynomial from Z[x] that is irreducible ov...
 17.4E: Suppose that If r is rational and x – r divides f(x), show that r i...
 17.5E: Let F be a field and let a be a nonzero element of F.a. If af(x) is...
 17.6E: Let F be a field and f(x) ? F[x]. Show that, as far as deciding upo...
 17.7E: Explain how the Mod p Irreducibility Test (Theorem 17.3) can be use...
 17.8E: Suppose that f(x)?Zp[x] and f(x) is irreducible over Zp, where p is...
 17.9E: Construct a field of order 25.
 17.10E: Construct a field of order 27.
 17.11E: Show that x3 + x2 + x + 1 is reducible over Q. Does this fact contr...
 17.12E: Determine which of the polynomials below is (are) irreducible over Q.
 17.13E: Show that x4 + 1 is irreducible over Q but reducible over R. (This ...
 17.14E: Show that x2 + x + 4 is irreducible over Z11.
 17.15E: Let f(x) = x3 + 6 ? Z7[x]. Write f(x) as a product of irreducible p...
 17.16E: Let f(x) = x3 + x2 + x + 1 ? Z2[x]. Write f(x) as a product of irre...
 17.17E: Let p be a prime.a. Show that the number of reducible polynomials o...
 17.18E: Let p be a prime.a. Determine the number of irreducible polynomials...
 17.19E: Show that for every prime p there exists a field of order p2.
 17.20E: Prove that, for every positive integer n, there are infinitely many...
 17.21E: Show that the field given in Example 11 in this chapter is isomorph...
 17.22E: Let f(x) ? Zp[x]. Prove that if f(x) has no factor of the form x2 +...
 17.23E: Find all monic irreducible polynomials of degree 2 over Z3.
 17.24E: Given that ? is not the zero of a nonzero polynomial with rational ...
 17.25E: Find all the zeros and their multiplicities of x5 + 4x4 + 4x3 – x2 ...
 17.26E: Find all zeros of f(x) = 3x2 + x + 4 over Z7 by substitution. Find ...
 17.27E: (Rational Root Theorem) and an ? 0. Prove that if r and s are relat...
 17.28E: Let F be a field and let p(x), a1(x), a2(x), . . . , ak(x) ? F[x], ...
 17.29E: Show that x4 + 1 is reducible over Zp for every prime p. (This exer...
 17.30E: If p is a prime, prove that xp–1 – xp–2 + xp–3 – ? ? ? – x + 1 is i...
 17.31E: Let F be a field and let p(x) be irreducible over F. If E is a fiel...
 17.32E: Prove that the ideal is prime in Z[x] but not maximal in Z[x].
 17.33E: Let F be a field and let p(x) be irreducible over F. Show that {a +...
 17.34E: Let F be a field and let f(x) be a polynomial in F[x] that is reduc...
 17.35E: Example 1 in this chapter shows the converse of Theorem 17.2 is not...
 17.36E: Suppose there is a real number r with the property that r + 1/r is ...
 17.37E: In the game of Monopoly, would the probabilities of landing on vari...
 17.38E: Carry out the analysis given in Example 12 for a pair of tetrahedro...
 17.39E: Suppose in Example 12 that we begin with n (n> 2) ordinary dice eac...
 17.40E: Show that one twosided die labeled with 1 and 4 and another 18 si...
Solutions for Chapter 17: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 17
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 17 have been answered, more than 36260 students have viewed full stepbystep solutions from this chapter. 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. Chapter 17 includes 41 full stepbystep solutions.

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

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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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