MTH 310 Lecture Notes Based on Hungerford, Abstract Algebra

177 Pages·2015·0.86 MB·English

MTH 310
Lecture Notes
Based on Hungerford, Abstract Algebra
Ulrich Meierfrankenfeld
Department of Mathematics
Michigan State University
East Lansing MI 48824
[email protected]
April 29, 2015
2
Contents
1 Set, Relations and Functions 5
1.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Relations and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 The Natural Numbers and Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Rings 29
2.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Elementary Properties of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 The General Associative, Commutative and Distributive Laws in Rings . . . . . . . . . 37
2.4 Divisibility and Congruence in Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Congruence in the ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6 Modular Arithmetic in Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 Subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.8 Units in Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.9 The Euclidean Algorithm for Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.10 Integral Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.11 Isomorphism and Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.12 Associates in commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3 Polynomial Rings 87
3.1 Addition and Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 Divisibility in F[x]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Polynomial function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.5 The Congruence Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.6 Congruence Class Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.7 F [α] when p is irreducible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
p
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4 CONTENTS
4 Ideals and Quotients 133
4.1 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.2 Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A Logic 143
A.1 Rules of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B Relations, Functions and Partitions 149
B.1 Equality of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.2 The inverse of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.3 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C Real numbers, integers and natural numbers 155
C.1 Definition of the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
C.2 Algebraic properties of the integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.3 Properties of the order on the integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.4 Properties of the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
D The Associative, Commutative and Distributive Laws 159
D.1 The General Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
D.2 The general commutative law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
D.3 The General Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
E Verifying Ring Axioms 165
F Constructing rings from given rings 167
F.1 Direct products of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
F.2 Matrix rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
F.3 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
G Cardinalities 175
G.1 Cardinalities of Finite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Chapter 1
Set, Relations and Functions
1.1 Logic
In this section we will provide an informal discussion of logic. A statement is a sentence which is
either true or false, for example
(1) 1+1=2
√
(2) 2 is a rational number.
(3) π is a real number.
(4) Exactly 1323 bald eagles were born in 2000 BC,
all are statements. Statement (1) and (3) are true. Statement (2) is false. Statement (4) is probably
false, but verification might be impossible. It nevertheless is a statement.
Let P and Q be statements.
“P and Q” is the statement that P is true and Q is true. We illustrate the statement P and Q
in the following truth table
P Q P and Q
T T T
T F F
F T F
F F F
“P or Q” is the statement that at least one of P and Q is true:
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6 CHAPTER 1. SET, RELATIONS AND FUNCTIONS
P Q P or Q
T T T
T F T
F T T
F F F
So “P or Q” is false exactly when both P and Q are false.
“not-P” (pronounced ‘not P’ or ‘negation of P’) is the statement that P is false:
P not-P
T F
F T
So not-P is true if P is false. And not-P is false if P is true.
“P (cid:212)⇒Q” (pronounced “P implies Q”) is the statement “ If P is true, then Q is true”:
P Q P (cid:212)⇒Q
T T T
T F F
F T T
F F T
Note here that if P is true, then “P (cid:212)⇒Q ” is true if and only if Q is true. But if P is false, then
“P (cid:212)⇒Q” is true, regardless whether Q is true or false. Consider the statement “ Q or not-P” :
P Q not-P Q or not-P
T T F T
T F F F
F T T T
F F T T
(∗) ”Q or not-P” is true if and only ”P (cid:212)⇒Q” is true.
1.1. LOGIC 7
This shows that one can express the logical operator “(cid:212)⇒” in terms of the operators ” not-”
and “or”.
“P ⇐⇒Q” (pronounced “P is equivalent to Q”) is the statement that P is true if and only if Q
is true.:
P Q P ⇐⇒Q
T T T
T F F
F T F
F F T
So P ⇐⇒Q is true if either both P and Q are true, or both P and Q are false. Hence
(∗∗) ”P ⇐⇒Q” is true if and only ”(P and Q) or (not-P and not-Q)” is true.
To show that P and Q are equivalent one often proves that P implies Q and that Q implies P.
Indeed the truth table
P Q P (cid:212)⇒Q Q(cid:212)⇒P (P (cid:212)⇒Q) and(Q(cid:212)⇒P)
T T T T T
T F F T F
F T T F F
F F F T T
shows that
(∗∗∗) ”P ⇐⇒Q” is true if and only ”(P (cid:212)⇒Q) and (Q(cid:212)⇒P)” is true.
Often, ratherthanshowingthatastatementistrue, oneshowsthatthenegationofthestatement
is false (This is called a proof by contradiction). To do this it is important to be able to determine
the negation of statement. The negation of not-P is P:
P not-P not-(not-P)
T F T
F T F
The negation of ”P and Q” is ”not-P or not-Q”:
8 CHAPTER 1. SET, RELATIONS AND FUNCTIONS
P Q P and Q not-(P and Q) not-P not-Q not-P or not-Q
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T F T
The negation of ”P or Q” is ”not-P and not-Q”:
P Q P or Q not-(P or Q) not-P not-Q not-P and not-Q
T T T F F F F
T F T F F T F
F T T F T F F
F F F T T F T
The statement “not-Q (cid:212)⇒ not-P” is called the contrapositive of the statement “P (cid:212)⇒ Q”. It
is equivalent to the statement “P (cid:212)⇒Q”:
P Q P (cid:212)⇒Q not-Q not-P not-Q(cid:212)⇒not-P
T T T F F T
T F F T F F
F T T F T T
F F T T T T
The statement “ not-P ⇐⇒not-Q” is called the contrapositive of the statement “P ⇐⇒Q”. It
is equivalent to the statement “P ⇐⇒Q”:
P Q P ⇐⇒Q not-P not-Q not-P ⇐⇒not-Q
T T T F F T
T F F F T F
F T F T F F
F F T T T T
1.1. LOGIC 9
The statement “Q (cid:212)⇒ P” is called the converse of the statement “ P (cid:212)⇒ Q”. In general the
converse is not equivalent to the original statement. For example the statement if x=0 then x is an
even integer is true. But the converse (if x is an even integer, then x=0) is not true.
Theorem 1.1.1 (Principal of Substitution). Let Φ(x) be formula involving a variable x. For an
object d let Φ(d) be the formula obtained from Φ(x) by replacing all occurrences of x by d. If a and
b are objects with a=b, then Φ(a)=Φ(b).
Proof. This should be self evident. For an actual proof and the definition of a formula consult your
favorite logic book.
Example 1.1.2. Let Φ(x)=x2+3⋅x+4.
If a=2, then
a2+3⋅a+4=22+3⋅2+4
Notation 1.1.3. Let P(x) be a statement involving the variable x.
(a) “for all x ∶ P(x)” is the statement that for all objects a the statements P(a) is true. Instead
of “for all x ∶ P(x)” we will also use “∀x ∶ P(x)”, ”P(x) is true for all x”, “P(x) holds for
all x” or similar phrases.
(b) ‘there exists x∶P(x)” is the statement there exists an object a such that the statements P(a)
is true. Instead of “there exists x∶P(x)” we will also use “∃x∶P(x)”, ”P(x) is true for some
x”, “There exists x with P(x)” or similar phrases.
Example 1.1.4. “for all x∶x+x=2x” is a true statement.
“for all x∶x2 =2” is a false statement.
“there exists x∶x2 =2” is a true statement.
“∃x∶x2 =2 and x is an integer” is false statement
Notation 1.1.5. Let P(x) be a statement involving the variable x.
(a) “There exists at most one x∶P(x)” is the statement
for all x∶ for all y ∶ P(x) and P(y) (cid:212)⇒ x=y
(b) “There exists a unique x∶P(x)” is the statement
there exists x∶ for all y ∶ P(y) ⇐⇒ y =x
10 CHAPTER 1. SET, RELATIONS AND FUNCTIONS
Example 1.1.6. “There exists at most one x∶(x2 =1 and x is a real number)” is false since 12 =1
and (−1)2 =1, but 1≠−1.
“There exists a unique x ∶ (x3 = −1 and x is a real number)” is true since x = −1 is the only
element in R with x3 =1.
“There exists at most one x ∶ (x2 = −1 and x is a real number)” is true, since there does not
exist any element x in R with x2 =−1.
“There exists a unique x ∶ (x2 = −1 and x is a real number)” is false, since there does not exist
any element x in R with x2 =−1.
Theorem 1.1.7. Let P(x) be a statement involving the variable x. Then
( there exists x∶P(x)) and (there exists at most onex∶P(x))
if and only if
there exists a uniquex∶P(x)
Proof. Consult A.1.2 in the appendix for the proof.
Exercises 1.1:
#1. Convince yourself that each of the statement in A.1.1 are true.
#2. Use a truth table to verify the statements LR 17, LR 26, LR 27 and LR 28 in A.1.1.
1.2 Sets
First of all any set is a collection of objects.
For example
Z∶={...,−4,−3,−2,−1,−0,1,2,3,4,...}
is the set of integers. If S is a set and x an object we write x∈S if x is a member of S and x∉S if
x is not a member of S. In particular,
(∗) For all x exactly one of x∈S and x∉S holds.
In other words:
for all x∶ x∉S ⇐⇒ not-(x∈S)
Not all collections of objects are sets. Suppose for example that the collection B of all sets is a
set. Then B ∈B. This is rather strange, but by itself not a contradiction. So lets make this example
a little bit more complicated. We call a set S nice if S ∉ S. Let D be the collection of all nice sets
and suppose D is a set.
Is D a nice set?