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
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