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MTH 513 Homework April 3, 2019

1. In your own words, state (explain? Prove?) why

(a) if R is a commutative ring with 1, then R[x], the
polynomial ring, is also a commutative ring.

(b) if R is an integral domain, then R[x] is also an integral
domain.

2. Show that for the numbers 50 and 20, 50 = 2 × 20 + 10 is
the unique representation of the

form 50 = × 20 + where are integers. (See the uniqueness part
of the proof of

Theorem 3.2 on pages 31-32.)

3. = 60, = 74.

(a) Find integers such that gcd( , ) = + .

(b) Are these integers unique? If so, why? If not, find
another pair of integers

′ ′ ℎ ℎ gcd( , ) = ′ + ′ .

3. Can a fifth-degree polynomial over the real numbers (i.e.
with coefficients in R) be

irreducible? If so, give an example. If not, explain why not.
Graph your polynomial.

4. Can a sixth-degree polynomial over the real numbers (i.e.
with coefficients in R) be

irreducible? If so, give an example. If not, explain why not.
Graph your polynomial function.

5. Do the following exercises that appear on the attached
sheets.

Page 93/ 1, 2, 3, 5, 6(i), (ii), 7, 8

An undergraduate course in

Abstract Algebra

Course notes for MATH3002 Rings and Fields

Robert Howlett

An undergraduate course in

Abstract Algebra

by

Robert Howlett

typesetting by TEX

Contents

Foreword

Chapter 0:

§0a

§0b

§0c

§0d

The integers

1

2

3

4

7

7

8

9

17

17

18

21

25

30

Quotients of the ring of integers

Equivalence relations

Congruence relations on the integers

The ring of integers modulo n

Properties of the ring of integers modulo n

Chapter 5:

§5a

§5b

§5c

§5d

Introduction to rings

Two basic properties of the integers

The greatest common divisor of two integers

Factorization into primes

Chapter 4:

§4a

§4b

§4c

§4d

Ruler and compass constructions

Operations on sets

The basic definitions

Two ways of forming rings

Trivial properties of rings

Chapter 3:

§3a

§3b

§3c

1

Three problems

Some examples of constructions

Constructible numbers

Chapter 2:

§2a

§2b

§2c

§2d

Prerequisites

Concerning notation

Concerning functions

Concerning vector spaces

Some very obvious things about proofs

Chapter 1:

§1a

§1b

§1c

v

30

33

38

42

42

44

45

48

Some Ring Theory

52

Subrings and subfields

Homomorphisms

Ideals

The characteristic of a ring

52

57

62

64

iii

Chapter 6:

§6a

§6b

§6c

§6d

§6e

§6f

§6g

§6h

§6i

§6j

§6k

More Ring Theory

More on homomorphisms

More on ideals

Congruence modulo an ideal

Quotient rings

The Fundamental Homomorphism Theorem

Chapter 8:

§8a

§8b

§8c

§8d

§8e

§8f

§8g

§8h

71

Definitions

Addition and multiplication of polynomials

Constant polynomials

Polynomial functions

Evaluation homomorphisms

The division algorithm for polynomials over a field

The Euclidean Algorithm

Irreducible polynomials

Some examples

Factorization of polynomials

Irreducibility over the rationals

Chapter 7:

§7a

§7b

§7c

§7d

§7e

Polynomials

Field Extensions

71

73

75

77

77

79

81

85

86

88

89

96

96

99

101

102

105

111

Ideals in polynomial rings

Quotient rings of polynomial rings

Fields as quotient rings of polynomial rings

Field extensions and vector spaces

Extensions of extensions

Algebraic and transcendental elements

Ruler and compass constructions revisited

Finite fields

Index of notation

Index of examples

iv

111

112

117

119

120

122

125

127

134

135

Foreword. . .

The purpose of this book is to complement the lectures and
thereby decrease,

but not eliminate, the necessity of taking lecture notes.
Reading the appropriate sections of the book before each lecture
should enable you to understand

the lecture as it is being given, provided you concentrate!
This is particularly

important in this course because, as theoretical machinery is
developed, the

lectures depend more and more heavily upon previous lectures,
and students

who fail to thoroughly learn the new concepts as they are
introduced soon

become completely lost.

???

Proofs of the theorems are an important part of this

course. You cannot expect to do third year Pure Mathematics

without coming to grips with proofs. Mathematics is about
proving

theorems. You will be required to know proofs of theorems for
the

exam.

???

It is the material dealt with in the lectures, not this book,
which defines the

syllabus of the course. The book is only intended to assist,
and how much

overlap there is with the course depends on the whim of the
lecturer. There

will certainly be things which are in the lectures and not in
the book, and

vice versa. The lecturer will probably dwell upon topics
which are giving

students trouble, and omit other topics. However, the book
will still provide

a reasonable guide to the course.

v

0

Prerequisites

Students will be assumed to be familiar with the material
mentioned in this

preliminary chapter. Anyone who is not should inform the
lecturer forthwith.

§0a

Concerning notation

When reading or writing mathematics you should always
remember that the

mathematical symbols which are used are simply abbreviations
for words.

Mechanically replacing the symbols by the words they
represent should result

in grammatically correct and complete sentences. The meanings
of a few

commonly used symbols are given in the following table.

Symbols

{ … | … }

=

∈

>

To be read as

the set of all . . . such that . . .

is

in or is in

greater than or is greater than

Thus for example the following sequence of symbols

{x ∈ X | x > a} =

6 ∅

is an abbreviated way of writing the sentence

The set of all x in X such that x is greater than a is not
the empty set.

When reading mathematics you should mentally translate all
symbols in this

fashion. If you cannot do this and obtain meaningful
sentences, seek help

from your tutor. And make certain that, when you use
mathematical symbols

yourself, what you write can be translated into meaningful
sentences.

1

2

Chapter Zero: Prerequisites

§0b

Concerning functions

The terminology we use in connection with functions could
conceivably differ

from that to which you are accustomed; so a list of
definitions of the terms

we use is provided here.

• The notation ‘f : A → B’ (read ‘f , from A to B’) means
that f is a

function with domain A and codomain B. In other words, f is a
rule which

assigns to every element a of the set A an element in the set
B denoted

by ‘f (a)’.

• A map is the same thing as a function. The term mapping is
also used.

• A function f : A → B is said to be injective (or
one-to-one) if and only

if no two distinct elements of A yield the same element of B.
In other words,

f is injective if and only if for all a1 , a2 ∈ A, if f (a1 )
= f (a2 ) then a1 = a2 .

• A function f : A → B is said to be surjective (or onto) if
and only if for

every element b of B there is an a in A such that f (a) = b.

• If a function is both injective and surjective we say that
it is bijective

(or a one-to-one correspondence).

• The image of a function f : A → B is the subset of B
consisting of all

elements obtained by applying f to elements of A. That is,

im f = { f (a) | a ∈ A }.

An alternative notation is ‘f (A)’ instead of ‘im f ’.
Clearly, f is surjective if

and only if im f = B.

• The notation ‘a 7→ b’ means ‘a maps to b’; in other words,
the function

involved assigns the element b to the element a. Thus ‘a 7→ b
under f ’ means

exactly the same as ‘f (a) = b’.

• If f : A → B is a function and C a subset of B then the
inverse image

or preimage of C is the subset of A

f −1 (C) = { a ∈ A | f (a) ∈ C }.

(The above line reads ‘f inverse of C, which is the set of
all a in A such that

f of a is in C.’ Alternatively, one could say ‘The inverse
image of C under

f ’ instead of ‘f inverse of C’.)

Chapter Zero: Prerequisites

§0c

Concerning vector spaces

Vector spaces enter into this course only briefly; the facts
we use are set out

in this section.

Associated with each vector space is a set of scalars. In the
common

and familiar examples this is R, the set of all real numbers,
but in general it

can be any field. (Fields are defined in Chapter 2.)

Let V be a vector space over F . (That is, F is the
associated field of

scalars.) Elements of V can be added and multiplied by
scalars:

(∗)

If v, w ∈ V and λ ∈ F then v + w, λv ∈ V .

These operations of addition and multiplication by scalars
satisfy the following properties:

(i) (u + v) + w = u + (v + w)

for all u, v, w ∈ V .

(ii) u + v = v + u

for all u, v ∈ V .

(iii) There exists an element 0 ∈ V such that v + 0 = v for
all v ∈ V .

(iv) For each v ∈ V there exists an element −v ∈ V such that
v + (−v) = 0.

(v) λ(µv) = (λµ)v

for all λ, µ ∈ F and all v ∈ V .

(vi) 1v = v

for all v ∈ V .

(vii) λ(v + w) = λv + λw

for all λ ∈ F and all v, w ∈ V .

(viii) (λ + µ)v = λv + µv

for all λ, µ ∈ F and all v ∈ V .

The properties listed above are in fact the vector space
axioms; thus in

order to prove that a set V is a vector space over a field F
one has only to

check that (∗) and (i)–(viii) are satisfied.

Let V be a vector space over F and let v1 , v2 , . . . vn ∈ V
. The elements

v1 , v2 , . . . vn are said to be linearly independent if the
following statement is

true:

If λ1 , λ2 , . . . , λn ∈ F and λ1 v1 + λ2 v2 + · · · + λn vn
= 0

then λ1 = 0, λ2 = 0, . . . , λn = 0.

The elements v1 , v2 , . . . vn are said to span the space V
if the following

statement is true:

For every v ∈ V there exist λ1 , λ2 , . . . , λn ∈ F

such that v = λ1 v1 + λ2 v2 + · · · + λn vn .

A basis of a vector space V is a finite subset of V whose
elements are linearly

independent and span V .

3

4

Chapter Zero: Prerequisites

We can now state the only theorem of vector space theory
which is used

in this course.

0.1 Theorem If a vector space V has a basis then any two
bases of V will

have the same number of elements.

Comment …

0.1.1

If V has a basis then the dimension of V is by definition the

number of elements in a basis.

…

§0d

Some very obvious things about proofs

When trying to prove something, the logical structure of what
you are trying to prove determines the logical structure of the
proof. The following

observations seem trivial, yet they are often ignored.

• To prove a statement of the form

If p then q

your first line should be

Assume that p is true

and your last line

Therefore q is true.

• The statement

p if and only if q

is logically equivalent to

If p then q and if q then p,

and so the proof of such a statement involves first assuming
p and proving q,

then assuming q and proving p.

• To prove a statement of the form

All xxxx’s are yyyy’s,

the first line of your proof should be

Let a be an xxxx

and the last line should be

Therefore a is a yyyy.

(The second line could very well involve invoking the
definition of ‘xxxx’

or some theorem about xxxx’s to determine things about a;
similarly the

second to last line might correspond to the definition of
‘yyyy’.)

Chapter Zero: Prerequisites

When trying to construct a proof it is sometimes useful to
assume

the opposite of the thing you are trying to prove, with a
view to obtaining

a contradiction. This technique is known as “indirect proof”
(or “proof

by contradiction”). The idea is that the conclusion c is a
consequence of

the hypotheses h1 , h2 , . . . , if and only if the negation
of c is incompatible

with h1 , h2 , . . . . Hence we may assume the negation of c
as an extra

hypothesis, along with h1 , h2 etc., and the task is then to
show that the

hypotheses contradict each other. Note, however, that
although indirect

proof is a legitimate method of proof in all situations, it
is not a good policy

to always use indirect proof as a matter of course. Most
proofs are naturally

expressed as direct proofs, and to recast them as indirect
proofs may make

them more complicated than necessary.

Examples

#1 Suppose that you wish to prove that a function λ: X → Y is
injective.

Consult the definition of injective. You are trying to prove
the following

statement:

For all x1 , x2 ∈ X, if λ(x1 ) = λ(x2 ) then x1 = x2 .

So the first two lines of your proof should be as follows:

Let x1 , x2 ∈ X.

Assume that λ(x1 ) = λ(x2 ).

Then you will presumably consult the definition of the
function λ to derive

consequences of λ(x1 ) = λ(x2 ), and eventually you will
reach the final line

Therefore x1 = x2 .

#2 Suppose you wish to prove that λ: X → Y is surjective.
That is, you

wish to prove

For every y ∈ Y there exists x ∈ X with λ(x) = y.

Your first line must be

Let y be an arbitrary element of Y .

Somewhere in the middle of the proof you will have to somehow
define an

element x of the set X (the definition of x is bound to
involve y in some

way), and the last line of your proof has to be

Therefore λ(x) = y.

5

6

Chapter Zero: Prerequisites

#3 Suppose that A and B are sets, and you wish to prove that
A ⊆ B.

(That is, A is a subset of or equal to B.) By definition the
statement ‘A ⊆ B’

is logically equivalent to

All elements of A are elements of B.

So your first line should be

Let x ∈ A

and your last line should be

Therefore x ∈ B.

#4 Suppose that you wish to prove that A = B, where A and B
are sets.

The following statements are all logically equivalent to ‘A =
B’:

(i) For all x, x ∈ A if and only if x ∈ B.

(ii) (For all x) (if x ∈ A then x ∈ B) and (if x ∈ B then x ∈ A)
.

(iii) All elements of A are elements of B and all elements of
B are elements

of A.

(iv) A ⊆ B and B ⊆ A.

You must do two proofs of the general form given in #3 above.

1

Ruler and compass constructions

Abstract algebra is essentially a tool for other branches of
mathematics.

Many problems can be clarified and solved by identifying
underlying structure and focussing attention on it to the exclusion
of peripheral information

which may only serve to confuse. Moreover, common underlying
structures

sometimes occur in widely varying contexts, and are more
easily identifiable

for having been previously studied in their own right. In
this course we shall

illustrate this idea by taking three classical geometrical
problems, translating them into algebraic problems, and then using
the techniques of modern

abstract algebra to investigate them.

§1a

Three problems

Geometrical problems arose very early in the history of
civilization, presumably because of their relevance to architecture
and surveying. The most

basic and readily available geometrical tools are ruler and
compass, for constructing straight lines and circles; thus it is
natural to ask what geometrical

problems can be solved with these tools.†

It is said that the citizens of Delos in ancient Greece, when
in the

grips of a plague, consulted an oracle for advice. They were
told that a god

was displeased with their cubical altar stone, which should
be immediately

replaced by one double the size. The Delians doubled the
length, breadth

and depth of their altar; however, this increased its volume
eightfold, and

the enraged god worsened the plague.

Although some historians dispute the authenticity of this
story, the

so-called “Delian problem”

† Note that the ruler is assumed to be unmarked; that is, it
is not a measuring

device but simply an instrument for ruling lines.

7

8

Chapter One: Ruler and compass constructions

(1) Given a cube, construct another cube with double the
volume

is one of the most celebrated problems of ancient
mathematics. There are

two other classical problems of similar stature:

(2) Construct a square with the same area as a given circle

(3) Trisect a given angle.

In this course we will investigate whether problems (1), (2)
and (3) can be

solved by ruler and compass constructions. It turns out that
they cannot.

We should comment, however, that although the ancient
mathematicians were unable to prove that these problems were
insoluble by ruler and

compass, they did solve them by using curves other than
circles and straight

lines.

§1b

Some examples of constructions

Before trying to prove that some things cannot be done with
ruler and compass, we need to investigate what can be done with
those tools. Much of

what follows may be familiar to you already.

#1 Given straight lines AB and AC intersecting at A the angle
BAC can

be bisected, as follows. Draw a circle centred at A, and let
X, Y be the

points where this circle meets AB, AC. Draw circles of equal
radii centred

at X and Y , and let T be a point of intersection of these
circles. (The radius

must be chosen large enough so that the circles intersect.)
Then AT bisects

the given angle BAC.

#2 Given lines AB and AC intersecting at A and a line P Q,
the angle

BAC can be copied at P , as follows. Draw congruent circles
CA , CP centred

at A and P . Let CA intersect AB at X and AC at Y , and let
CP intersect

P Q at V . Draw a circle with centre V and radius equal to XY
, and let T

be a point of intersection of this circle and CP . Then the
angle T P Q equals

the angle BAC.

#3 Given a point A and a line P Q, one can draw a line
through A parallel

to P Q. Simply draw any line through A intersecting P Q at
some point X,

and then copy the angle AXQ at the point A.

#4 Given a line AB one can construct a point T such that the
angle T AB

equals π3 radians (60 degrees). Simply choose T to be a point
of intersection

of the circle centred at A and passing through B and the
circle centred at B

and passing through A.

Chapter One: Ruler and compass constructions

#5 Given line segments of lengths r, s and t one can
construct a line

segment of length rt/s, as follows. Draw distinct lines AP ,
AQ intersecting

at A and draw circles Cr , Cs and Ct of radii r, s and t
centred at A. Let Cr

intersect AP at B and let Cs , Ct intersect AQ at X, Y . Draw
a line through

Y parallel to XB, and let C be the point at which it
intersects AP . Then

AC has the required length.

#6 There are simple constructions for angles equal to the sum
and difference of two given angles, lengths equal to the sum and
difference of two

given lengths, and for a/n and na, where a is a given length
and n a given

positive integer. See the exercises at the end of the
chapter.

#7 Given line segments of lengths √

a and b, where a ≥ b, it is possible

to construct a line segment of length ab, as follows. First,
construct line

segments of lengths r1 = 21 (a + b) and r2 = 21 (a − b), and
draw circles of

radii r1 and r2 with the same centre O. Draw a line through O
intersecting

the smaller circle at P , and draw a line through P
perpendicular to OP . (A

right-angle can be constructed, for instance, by constructing
an angle of π3 ,

bisecting it, and adding on another angle of π3 .) Let this
perpendicular meet

the large circle at Q. Then P Q has the required length.

Further ruler and compass constructions are dealt with in the
exercises.

§1c

Constructible numbers

Consider the Delian Problem once more: we are given a cube
and wish to

double its volume. We may as well choose our units of length
so that the

given cube has sides

√ of length one. Then our problem is to construct a line

segment of length 3 2. The other problems can be stated
similarly. A circle of

unit radius has area π; to√construct a square of this area
one must construct

a line segment of length π. A right-angled triangle with unit
hypotenuse

and an

angle θ has other sides cos θ and sin θ; to trisect θ one
must construct

θ

cos 3 . So the problems become:

√

(1) Given a unit line segment, construct one of length π.

√

(2) Given a unit line segment, construct one of length 3 2.

(3) Given line segments of lengths 1 and cos θ, construct one
of length

cos θ3 .

9

10

Chapter One: Ruler and compass constructions

To show that Problem 3 cannot be solved by ruler and compass,
it will

be sufficient to sh …

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