 44.44.1: Prove that in Example 44.1, (Y(n) = n for every positive integer n.
 44.44.2: Prove that the set Aut(E) .of all automorphisms of a field E is a g...
 44.44.3: Prove the statement in (44.1).
 44.44.4: Prove the statement in (44.2).
 44.44.5: What is Gal(Q(.j2,.J3) : Q)? (Find the elements and construct a Cay...
 44.44.6: Prove that Gal(E/F) is a subgroup of Aut(E).
 44.44.7: Prove that E H, as defined in Theorem 44.1, is a subfield of E
 44.44.8: In Example 44.4, verify that Gal(Q(w, !fi)/Q( !fi is {I, a}.
 44.44.9: In Example 44.4, what is the fixed field of (a)?
 44.44.10: Verify that Gal(Q(0)/Q) "" Z2.
 44.44.11: Verify that Gal(Q( YS)/Q) has order one.
 44.44.12: (a) Show that the splitting field of X4  5 over Q is Q(~, i).(b) S...
 44.44.13: Verify statements (i), (ii), and (iii) in Example 44.3.
Solutions for Chapter 44: FUNDAMENTAL THEOREM: INTRODUCTION
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 44: FUNDAMENTAL THEOREM: INTRODUCTION
Get Full SolutionsChapter 44: FUNDAMENTAL THEOREM: INTRODUCTION includes 13 full stepbystep 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. This expansive textbook survival guide covers the following chapters and their solutions. Since 13 problems in chapter 44: FUNDAMENTAL THEOREM: INTRODUCTION have been answered, more than 8320 students have viewed full stepbystep solutions from this chapter.

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.

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

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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

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

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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 .

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.