- 36.36.1: Use the Euclidean Algorithm to compute the greatest common divisors...
- 36.36.2: Use the Euclidean Algorithm to compute the greatest common divisors...
- 36.36.3: Use the Euclidean Algorithm to compute the greatest common divisors...
- 36.36.4: Use the Euclidean Algorithm to compute the greatest common divisors...
- 36.36.5: Determine the monic associate of 2x3 - x + 1 E Q[x 1.
- 36.36.6: Determine the monic associate of 1 + x - ix2 E iC[x1.
- 36.36.7: Determine the monic associate of 2x5 - 3x2 + 1 E Z7[x1.
- 36.36.8: Determine the monic associate of2x5 - 3x2 + 1 E Z5[x1.
- 36.36.9: Verify that x3 - 3 E Z7[X] is irreducible.
- 36.36.10: Verify that X4 + x 2 + 1 E Z5[X] is reducible.
- 36.36.11: Write x 3 + 3x2 + 3x + 4 E Z5 [x] as a product of irreducible polyn...
- 36.36.12: Write x 5 + X4 + x 2 + 2x E Zl [x] as a product of irreducible poly...
- 36.36.13: Prove that the polynomial lAx) in the proof of Theorem 36.1 satisfi...
- 36.36.14: Prove the uniqueness of d(x) in Theorem 36.1
- 36.36.15: Prove Theorem 36.2. (The remark preceding it suggests the method.)
- 36.36.16: Explain why each nonzero polynomial has preCisely one monic polynom...
- 36.36.17: Write the proof of the second corollary of Theorem 36.2.
- 36.36.18: Complete the proof of the Unique Factorization Theorem
- 36.36.19: (a) Prove that (x - I) I I(x) in Z2[X] iff I(x) has an even number ...
- 36.36.20: (a) By counting the number of distinct possibilities for (x - alex ...
- 36.36.21: State and prove a theorem establishing the existence of a unique le...
- 36.36.22: (Eisenstein's irreducibility criterion) Assume that p is a prime, I...
- 36.36.23: Use Eisenstein's irreducibility criterion ( 36.22) to show that eac...
- 36.36.24: Use Eisenstein's irreducibility criterion ( 36.22) to prove that if...
Solutions for Chapter 36: FACTORIZATION OF POLYNOMIALS
Full solutions for Modern Algebra: An Introduction | 6th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
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.
A sequence of steps intended to approach the desired solution.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Solvable system Ax = b.
The right side b is in the column space of A.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.