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Math 561 H
Homework 3 Solutions
Fall 2011
Drew Armstrong
Group Problems.
1. Let G be a group. Given a G dene the centralizer Z (a) := cfw_b G : ab = ba. Prove that
Z (a) G. For which a G is Z (a) = G?
Proof. To show closure, let b, c Z (a). That is, suppo
Math 561 H
Homework 2 Solutions
Fall 2011
Drew Armstrong
1. Let : G H be a homomorphism of groups. Prove that im is a subgroup of H .
Proof. First we show that im is closed. To see this, suppose that x, y im , so there exist
a, b G such that (a) = x and (
Math 561 H
Exam 3 Wed Nov 30
Fall 2011
Drew Armstrong
There are 3 problems and 5 pages. This is a closed book test. Any student caught cheating
will receive a score of zero.
1. Suppose that a group G acts on a set X by homomorphism : G Aut(X ), and
dene a
Math 561 H
Exam 2 Mon Oct 31
Fall 2011
Drew Armstrong
There are 3 problems with a total of 9 sections. This is a closed book test. Any student
caught cheating will receive a score of zero. In any of the 9 sections, you may assume the
results from the othe
Math 561 H
Exam 1 Wed Sep 28
Fall 2011
Drew Armstrong
There are 4 problems and 4 pages. This is a closed book test. Any student caught cheating
will receive a score of zero. The problems are (mostly) cumulative. In any problem, you
may assume the results
Math 561 H
Homework 4
Fall 2011
Drew Armstrong
1. Dene the ring of quaternions H := cfw_a1 + bi + cj + dk : a, b, c, d R, with the relations
1 = 1 and i2 = j2 = k2 = ijk = 1. Dene the quaternion absolute value by
|a1 + bi + cj + dk|2 := a2 + b2 + c2 + d2
Math 561 H
Homework 5
Fall 2011
Drew Armstrong
1. We saw in class that any element of the orthogonal group O(2) has the form
R :=
cos sin
sin cos
or
F :=
cos sin
.
sin cos
The matrix R (with determinant 1) rotates the plane around 0 counterclockwise b
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Homework 3 (complete)
September 12, 2013
Due on September 19th.
1-Read sections 2.7, 2.8, 2.9
2-Exercises: 2.7.1, 2.8.2 , 2.8.3, 2.8.5, 2.8.6, 2.8.8, 2.8.10, 2.8.13, 2.9.2, 2.9.4, 2.9.4, 2.9.6
Problem: Prove that the center Z(GLn (C) = cfw_I with = 0 taki
Homework 9 (complete)
November 13, 2013
Due December 3rd.
1-Read sections 5.1, 6.1, 6.2, 6.3, 6.4
2- The midterm will cover what we did from chapters 3,4,5 and 6.
2-Exercises: 5.1.1, 5.1.4, 6.3.2, 6.3.3, 6.3.6, 6.4.1, 6.4.2, 6.4.3
1
Homework 1 (complete)
August 29, 2013
Due on September 5th.
1-Read sections 2.1, 2.2
2-Exercises: 2.1.1, 2.1.2, 2.2.1, 2.2.3, 2.2.6
3- Show that if A and B are nxn matrices such that AB=I, then BA=I.
1
Math 461 F Quadratic Field Extensions
Spring 2011 Drew Armstrong
Let F be a field and let c F be an element such that c F . (This notation means that the equation x2 - c = 0 has no solution in F .) In this case we can define a new, bigger number system F
Math 461 F Fundamental Theorem of Algebra
Spring 2011 Drew Armstrong
When we proved the impossibility of the classical construction problems, we were interested in the existence of certain roots of polynomials. The flavor of what we did is contained in th
Math 461 F Impossible Constructions
Spring 2011 Drew Armstrong
In class we proved that 2 is not a rational number (the edge and diagonal of a square are incommensurable). This "crisis of incommensurables" forced the Greeks to base their mathematics on th
Math 461 F Homework 6
Reading. Section 6.4, 6.5 Book Problems. 6.4: 1, 4, 8. Additional Problems.
Spring 2011 Drew Armstrong
A.1. Euler showed that every real polynomial of the form x4 + x2 + x + factors into two real quadratics. Use his result to prove t
Math 461 F Homework 5 Solutions
Problems.
Spring 2011 Drew Armstrong
A.1. Let f (x) = an xn + + a1 x + a0 R[x]. If n is even, with an > 0 and a0 < 0, prove that f (x) has at least two real roots. (Hint: Intermediate value theorem.) Consider the graph of f
Math 461 F Homework 5
Reading. Chapter 6 Problems.
Spring 2011 Drew Armstrong
A.1. Let f (x) = an xn + + a1 x + a0 R[x]. If n is even, with an > 0 and a0 < 0, prove that f (x) has at least two real roots. (Hint: Intermediate value theorem.) A.2. Leibniz (
Math 461 F Homework 4 Solutions
Spring 2011 Drew Armstrong
A.1. Euclid's Lemma. Suppose that a divides bc for a, b, c Z with a and b coprime (i.e. they have no common factor except 1). Prove that a must divide c. (Hint: Since a and b are coprime, you may