 43.43.1: Prove that x 2  1 E Zdx] has four roots in Z12. Does this contradi...
 43.43.2: Construct a polynomial over Zs having 3 (that is, [3]) as a root of...
 43.43.3: Construct a polynomial over IC having i as a root of mUltiplicity t...
 43.43.4: There are eight polynomials of degree three over Z2. For each one, ...
 43.43.5: Prove that 12xl  3x + 2 has no rational root.
 43.43.6: Find all rational roots of each of the following polynomials over Q...
 43.43.7: Write each of the following polynomials over Q as a product of fact...
 43.43.8: (a) to (c). Repeat 43.7 using factors that are irreducible over R
 43.43.9: (a) to (c). Repeat 43.7 fusing factors that are irreducible over IC.
 43.43.10: There are nine monic polynomials of degree 2 over Zl. For each one,...
 43.43.11: Prove that if I(x) E lR(x] has an imaginary root of multiplicity tw...
 43.43.12: Give an example of a quadratic polynomial over IC that has an imagi...
 43.43.13: Give an example of a polynomial over Z:2 that is irreducible and of...
 43.43.14: Prove that an equivalent definition of splitting field results if c...
 43.43.15: Prove that the mapping II in Example 43.3 is an automorphism of IC.
 43.43.16: Assume that F is a field, I(x) E F[x], and E is an extension of F. ...
 43.43.17: Prove that if Q denotes the division ring of quartemions introduced...
 43.43.18: Assume that ao, ai, ... , a" are distinct elements of a field F, th...
Solutions for Chapter 43: ROOTS OF POLYNOMIALS
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 43: ROOTS OF POLYNOMIALS
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Since 18 problems in chapter 43: ROOTS OF POLYNOMIALS have been answered, more than 8340 students have viewed full stepbystep solutions from this chapter. Chapter 43: ROOTS OF POLYNOMIALS includes 18 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

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

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

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.