MATH 145 Algebra, Lecture Notes
by Stephen New
Chapter 0: Logic and Proof
1. Introduction
1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs
contained paradoxes, and these paradoxes could be used to prove statements that w
MATH 145 Algebra, Solutions to Assignment 9
1: Solve each of the following equations for z C. Express your answers in cartesian form.
(a) i z 2 + (2 + i) z + (7 + i) = 0
Solution: By the Quadratic Formula, the two solutions are
p
p
(2 + i) (2 + i)2 4i(7 +
Chapter 5. Complex Numbers
5.1 Definition: A complex number is a vector in R2 . The complex plane, denoted
by C, is the set of complex numbers:
x
2
C=R =
x R, y R .
y
0
1
0
x
0
In C we usually write 0 =
,1=
,i=
,x=
, iy = yi =
and
0
0
1
0
y
2. FACTORIZING IN INTEGRAL DOMAINS
5
Chapter 2
Factorizing In Integral Domains
Let R be an integral domain.
Definition 2.1. If r, s R and s = rt for some t R, then we say that r divides s.
This is written as r|s.
divides
r|s
Example 2.2.
1. If R = Z, this
4
subfield
F (1 , . . . , n )
1. BASIC PROPERTIES OF RINGS
Definition 1.13. A subset S of a field F is a subfield if S is a field with the same
addition and multiplication.
To check that S is a subfield, it is enough to check that for any a, b S, a + b, a
CONTENTS
iii
Contents
1 Basic Properties Of Rings
1
2 Factorizing In Integral Domains
5
3 Euclidean domains and principal ideal domains
11
4 Homomorphisms and factor rings
19
5 Field extensions
29
6 Ruler and Compass Constructions
33
7 Finite fields
43
MATH 135 Algebra, Solutions to the Final Exam, Fall 2009
[6]
1: (a) Let a0 = 1 and a1 = 3, and for n 2 let an = 3an1 2an2 1. Show that an = 2n + n for all n 0.
Solution: We claim that an = 2n +n for all n 0. When n = 0 we have an = a0 = 1 and 2n +n = 20 +
MATH 145 Algebra, Solutions to Assignment 6
1: For each of the following pairs (a, b), use the Euclidean Algorithm with Back-Substitution to find d = gcd(a, b)
and to find integers s and t such that as + bt = d.
(a) a = 759, b = 239
Solution: The Euclidea
MATH 145 Algebra, Solutions to Assignment 8
1: (a) Find 17458 mod 13.
Solution: Since 17 = 4 mod 13 we have 17458 = 4458 mod 13. By Fermats Little Theorem, the list of powers
of 4 modulo 13 repeats every 12 terms, and we have 458 = 2 mod 12, and so
17458
MATH 145 Algebra, Assignment 6
Due Fri Nov 13
1: For each of the following pairs (a, b), use the Euclidean Algorithm with Back-Substitution
to nd d = gcd(a, b) and to nd integers s and t such that as + bt = d.
(a) a = 759, b = 239
(b) a = 456, b = 1273
(c
MATH 145 Algebra, Assignment 5
Due Fri Oct 30
1: The axioms for a ring and for an ordered eld can be found on the following page.
(a) Let F be an ordered eld. Using only the 14 axioms which dene an ordered eld,
show that for all a, b, c F , if 0 a and b c
MATH 145 Algebra, Assignment 4
Not to hand in
In this assignment, a is a constant symbol; x, y and z are variable symbols; f , g and h are
function symbols with f unary and g and h binary; and r is a binary relation symbol.
1: Consider the interpretation
MATH 145 Algebra, Assignment 3
Due: Fri Oct 9
1: Make a derivation for each of the following equivalences. Provide justication at each step.
(a) (P Q) (P R) P R
(b) (P Q) R R P P
(c) (P (Q R) (P Q) (P Q) R
2: (a) Make a derivation for the valid argument (
MATH 145 Algebra, Assignment 2
Due: Fri Sept 25
1: Let S = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10. List all of the elements in each of the following sets.
(a) A = x S x is even or x is a multiple of 3
(b) B = x S if x is even then x is a multiple of 3
(c) C =
MATH 145 Algebra, Assignment 1
Due: Fri Sept 18
x
1
.
=
1+x
3 x
with 0 < x 1 for x in terms of y.
1: (a) Find all real numbers x such that
(b) Solve y = x +
1
x
2: (a) Find all ordered pairs of integers (x, y) such that xy = 6 + 2x.
(b) Find all ordered
MATH 145 Algebra, The Euclidean Algorithm with Back Substitution
Theorem: (The Euclidean Algorithm With Back-Substitution) Let a and b be integers
and let d = gcd(a, b). Then there exist integers s and t such that as + bt = d. The proof
provides explicit
MATH 145 Algebra (Advanced Level), Course Outline, Fall 2015
Lectures: Mon, Wed, Fri 12:30-1:20 in MC 2038, and Thurs 12:30-1:20 in MC 4059.
Instructor: Stephen New, tel: 35554, oce: MC 5419, hours: Mon, Wed, Thurs 1:30-2:30
Text: Integers, Polynomials an
1. BASIC PROPERTIES OF RINGS
1
Chapter 1
Basic Properties Of Rings
Definition 1.1. A ring R is a set with two binary operations, + and , satisfying:
ring
(1) (R, +) is an abelian group,
(2) R is closed under multiplication, and (ab)c = a(bc) for all a, b,
2
Z[ m]
Gaussian integers
1. BASIC PROPERTIES OF RINGS
Definition 1.5. Let d be an integer which is not a square. Define Z[ m] =
cfw_a + b m | a, b Z.
Call Z[ 1] = a + b 1, a, b Z the ring of Gaussian integers.
Proposition 1.6. Z[ d] is a ring. Moreove
page 1
Name (print):
Signature:
ID Number:
MATH 145, Algebra (Advanced Level)
Midterm Test, Fall Term, 2015
University of Waterloo
Instructor: Stephen New
Date:
October 19th, 2015
Time:
7:00-8:50 pm
Place:
MC 4042, 4058, 4063
Instructions:
1. Place your n
MATH 145 Algebra, Assignment 8
Due Fri Nov 27
1: (a) Find 17458 mod 13.
54
(b) Find 4738
mod 11.
1
57
5958
(c) Find 60
mod 19
300
P k
(c) Find
k mod 7.
k=1
2: (a) Show that 21(4n7 + 7n3 + 10n) for all integers n.
(b) Find a positive integer k such that th
MATH 145 Algebra, Assignment 7
Not to hand in
1: (a) Solve the linear diophantine equation 385x 1183y = 294.
(b) Find all pairs of integers (x, y) with x 1000, y 1000 such that 726x 1578y = 324.
(c) A shopper spends $19.81 to buy some apples which cost 35
1. BASIC PROPERTIES OF RINGS
3
Notation. If R is an integral domain (or any ring), then R[x] denotes the set of
polynomials in x with coefficients from R with usual addition and multiplication.
Clearly R[x] is a commutative ring.
R[x]
Proposition 1.9. If
6
2. FACTORIZING IN INTEGRAL DOMAINS
In general, R is not the same as R \ cfw_0.
Example 2.6 (of units).
1. Z = cfw_1.
2. Clearly, an integral domain F is a field iff F = F \ cfw_0.
3. Z[i] = cfw_1, 1, i, i: Suppose that a+bi Z[i] is a unit, so (a+bi)(c+d
MATH 145 Algebra, Solutions to Assignment 2
1: Let S = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10. List all of the elements in each of the following sets.
(a) A = x S x is even or x is a multiple of 3
Solution: We have
A = x S x is even x S x is a multiple
MATH 145 Algebra, Solutions to Assignment 4
In this assignment, a is a constant symbol; x, y and z are variable symbols; f , g and h are function symbols
with f unary and g and h binary; and r is a binary relation symbol.
1: Consider the interpretation wh