CHAPTER 8. POLYNOMIALS
110
Week 11
Polynomials over a Ring (Continued)
Recall: If () 6= 0, then the degree of (), abbreviated deg (), is the largest integer
such that the coecient of is not zero, and is called the leading coecient of
(). The term 0 is ca
Math 212, Assignment 2
February 2, 2017
Due Thursday, February 2, 2017
The graded questions will come from the first seven
1. For two integers d, n, we say that d is a divisor of n if there is another
integer i such that di = n. We say that d is a common
Math 212, Assignment 2
January 20, 2017
Due Thursday, February 2, 2017
The graded questions will come from the first seven
1. For two integers d, n, we say that d is a divisor of n if there is another
integer i such that di = n. We say that d is a common
Math 212, Assignment 1
January 9, 2017
Due Thursday, January 19, 2016
The exercises to be graded will come from the first five problems.
1. For each of the following sets, determine if the given operation is a
binary operation or not. Explain your answers
Chapter 3
Groups
Week 3
This is where the fun really starts!
3.1
Denition of a Group
An algebraic structure, or algebraic system, is a nonempty set in which at least one
equivelence relation (equality) and at least one binary operation are dened. The simp
CHAPTER 3. GROUPS
55
Week 6
Denition 3.25 Isomorphism, Automorphism
Let ( ) and (0 0 ) be groups. A mapping : 0 is an isomorphism from to 0
if
1. is a 11 correspondence from onto 0 , and
2. ( ) = () 0 () for all .
 cfw_z

cfw_z
in
0 in 0
If there e
CHAPTER 4. MORE ON GROUPS
67
Week 7
Denition 4.6 Conjugating Elements
If and are elements of the group , the conjugate of by is the element 1 . Any
element = 1 for some is called a conjugate of .
Note: If is abelian, then this concept is trivial and is th
CHAPTER 3. GROUPS
49
Week 5
Cyclic Groups (Continued)
What do generators of subgroups of nite cyclic groups look like?
Theorem 3.22 Generators of Subgroups of a Cyclic Group
Let = hi be a nite cyclic group of order . For any integer , the subgroup generat
CHAPTER 3. GROUPS
40
Week 4
Denition 3.11 Integral exponents
Let ( ) be a group. For any we dene nonnegative integral exponents by
0 = 1 =
and
+1 =
for any Z+ .
Negative integral exponents are dened by
= (1 )
for any Z+ .
The binary operation in abelia
CHAPTER 4. MORE ON GROUPS
80
Week 8
4.6
Quotient Groups
If C , then = for all there is no distinction between left and right cosets.
We simply refer to the cosets of in .
Theorem 4.21 Group of Cosets
If C , then the cosets of in form a group with respect
CHAPTER 8. POLYNOMIALS
119
Week 12
Theorem 8.21 Irreducible Factors
If () is an irreducible polynomial over the eld and ()[()()] in [], then
() () or ()().
Proof.
Assume that () is irreducible over and that ()[ ()()]; say
()() = ()()
(8.3.2)
for some
CHAPTER 7. REAL AND COMPLEX NUMBERS
98
Week 10
The Quaternions (Continued)
Recall: The set = cfw_( ) : R with equality, addition and multiplication
dened by
( ) = ( ) if and only if = = = = ;
( ) + ( ) = ( + + + + )
and
( )( ) = ( + + +
is a noncommutati
CHAPTER 5. RINGS, INTEGRAL DOMAINS AND FIELDS
89
Week 9
5.2
Integral Domains and Fields
Denition 5.14 Integral Domain
A ring is an integral domain if () is a commutative ring which () has a unity 6= 0
and () no zero divisors.
The condition that an integra
CHAPTER 1. FUNDAMENTALS
14
Week 2
Denition 1.38 Equivalence Class
Let be an equivalence relation on the nonempty set . For each , the set
[] = cfw_ :
is called the equivalence class containing .
Examples 1.38
1. Consider the equivalence relation in Examp
Math 212 Course Outline, Spring 2014
Web Page: www.math.uvic.ca. Look under course pages, then Math 212.
Days: Monday, Thursday Time: 13:00 14:20 Room: ELL 061
Instructor:
Dr. Kieka Mynhardt
O ce: DTB541
O ce Hours:
Math 212 students get preference
Monda