 21.1E: Prove Theorem 21.2 and Theorem 21.3.
 21.2E: Let E be the algebraic closure of F. Show that every polynomial in ...
 21.3E: Prove that is an algebraic extension of Q but not a finite extensio...
 21.4E: Let E be an algebraic extension of F. If every polynomial in F[x] s...
 21.5E: Suppose that F is a field and every irreducible polynomial in F[x] ...
 21.6E: Suppose that f (x) and g(x) are irreducible over F and that deg f (...
 21.7E: Let a and b belong to Q with b ? 0. Show that if and only if there ...
 21.8E: Find the degree and a basis for Find the degree and a basis for .
 21.9E: Suppose that E is an extension of F of prime degree. Show that, for...
 21.10E: Let a be a complex number that is algebraic over Q. Show that is al...
 21.11E: Suppose that E is an extension of F and a, b ? E. If a is algebraic...
 21.12E: Find an example of a field F and elements a and b from some extensi...
 21.13E: Let K be a field extension of F and let a ?K. Show that [F(a):F(a3)...
 21.14E: Find the minimal polynomial for
 21.15E: Let K be an extension of F. Suppose that E1 and E2 are contained in...
 21.16E: Find the minimal polynomial for
 21.17E: Let E be a finite extension of R. Use the fact that C is algebraica...
 21.18E: Suppose that [E:Q] = 2. Show that there is an integer d such that w...
 21.19E: Suppose that p(x) ? F[x] and E is a finite extension of F. If p(x) ...
 21.20E: Let E be an extension field of F. Show that [E:F] is finite if and ...
 21.21E: If ? and ? are real numbers and ? and ? are transcendental over Q, ...
 21.22E: Let f (x) be a nonconstant element of F[x]. If a belongs to some ex...
 21.23E: Let f (x) = ax2 + bx + c ? Q[x]. Find a primitive element for the s...
 21.24E: Find the splitting field for x4 + x2 + 2 over Z3.
 21.25E: Let f (x) ? F[x]. If deg f (x) = 2 and a is a zero of f (x) in some...
 21.26E: Let a be a complex zero of x2 + x + 1 over Q. Prove that
 21.27E: If F is a field and the multiplicative group of nonzero elements of...
 21.28E: Let a be a complex number that is algebraic over Q and let r be a r...
 21.29E: Prove that, if K is an extension field of F, then [K:F] = n if and ...
 21.30E: Let a be a positive real number and let n be an integer greater tha...
 21.31E: Let a and b belong to some extension field of F and let b be algebr...
 21.32E: Let f (x) and g(x) be irreducible polynomials over a field F and le...
 21.33E: Let ? be a zero of f(x) = x5 + 2x + 4 (see Example 8 in Chapter 17)...
 21.34E: Prove that
 21.35E: Let a and b be rational numbers. Show that
 21.36E: Let F, K, and L be fields with F ? K ? L. If L is a finite extensio...
 21.37E: Let F be a field and K a splitting field for some nonconstant poly...
 21.38E: Prove that C is not the splitting field of any polynomial in Q[x].
 21.39E: Prove that is not an element of Q(?).
 21.40E: Let . Prove that ? is not in Q(?).
 21.41E: Suppose that a is algebraic over a field F. Show that a and 1 + a–1...
 21.42E: Suppose K is an extension of F of degree n. Prove that K can be wri...
Solutions for Chapter 21: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 21
Get Full SolutionsContemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708. Chapter 21 includes 42 full stepbystep solutions. This 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. Since 42 problems in chapter 21 have been answered, more than 15460 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Column space C (A) =
space of all combinations of the columns of A.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

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.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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!

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.
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