- 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 two-sided die labeled with 1 and 4 and another 18- si...
Solutions for Chapter 17: Factorization of Polynomials
Full solutions for Contemporary Abstract Algebra | 8th Edition
ISBN: 9781133599708
Solutions for Chapter 17
24
4
Summary of Chapter 17: Factorization of Polynomials
In high school, students spend much time factoring polynomials and finding their zeros. In this chapter, we consider the same problems in a more abstract setting.
This expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 17: Factorization of Polynomials have been answered, more than 229266 students have viewed full step-by-step 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: Factorization of Polynomials includes 41 full step-by-step solutions.
Key Math Terms and definitions covered in this textbook
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Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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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.
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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.
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Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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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.
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Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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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).
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Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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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).
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Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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Solvable system Ax = b.
The right side b is in the column space of A.
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Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.