Concrete Algebra With a View Toward Abstract Algebra

387 Pages·2022·4.53 MB·English

by  Benjamin McKay

Benjamin McKay
Concrete Algebra
With a View Toward Abstract Algebra
December 4, 2022
ThisworkislicensedunderaCreativeCommonsAttribution-ShareAlike4.0UnportedLicense.
iii

Preface
WithmyfullphilosophicalrucksackIcanonlyclimbslowlyupthemoun-
tain of mathematics.
— Ludwig Wittgenstein
Culture and Value
These notes are from lectures given in 2015 at University College Cork. They aim
to explain the most concrete and fundamental aspects of algebra, in particular the
algebra of the integers and of polynomial functions of a single variable, grounded
by proofs using mathematical induction. It is impossible to learn mathematics by
reading a book like you would read a novel; you have to work through exercises and
calculate out examples. You should try all of the problems. More importantly, since
the purpose of this class is to give you a deeper feeling for elementary mathematics,
rather than rushing into advanced mathematics, you should reflect about how the
simple ideas in this book reshape your vision of algebra. Consider how you can use
your new perspective on elementary mathematics to help you some day guide other
students, especially children, with surer footing than the teachers who guided you.
v
vi
The temperature of Heaven can be rather accurately computed.
Our authority is Isaiah 30:26, “Moreover, the light of the Moon
shall be as the light of the Sun and the light of the Sun shall
be sevenfold, as the light of seven days.” Thus Heaven receives
from the Moon as much radiation as we do from the Sun, and
in addition 7×7=49 times as much as the Earth does from the
Sun, or 50 times in all. The light we receive from the Moon is
one1/10000ofthelightwereceivefromtheSun,sowecanignore
that.... The radiation falling on Heaven will heat it to the point
wheretheheatlostbyradiationisjustequaltotheheatreceivedby
radiation,i.e.,Heavenloses50timesasmuchheatastheEarthby
radiation. Using the Stefan-Boltzmann law for radiation, (H/E)
temperature of the earth (∼ 300K), gives H as 798K (525◦C).
The exact temperature of Hell cannot be computed.... [However]
Revelations 21:8 says “But the fearful, and unbelieving ...shall
havetheirpartinthelakewhichburnethwithfireandbrimstone.”
Alakeofmoltenbrimstonemeansthatitstemperaturemustbeat
or below the boiling point, 444.6◦C. We have, then, that Heaven,
at 525◦C is hotter than Hell at 445◦C.
— Applied Optics , vol. 11, A14, 1972
Inthesedaystheangeloftopologyandthedevilofabstractalgebra
fight for the soul of every individual discipline of mathematics.
— Hermann Weyl
Invariants,DukeMathematicalJournal5,1939,489–
502
— and so who are you, after all?
—Iampartofthepowerwhichforeverwillsevilandforeverworks
good.
— Goethe
Faust
This Book is not to be doubted.
— Quran , 2:1/2:6-2:10 The Cow
Contents
1 The integers 1
2 Mathematical induction 11
3 Greatest common divisors 21
4 Prime numbers 25
5 Modular arithmetic 29
6 The Chinese remainder theorem 37
7 Secret messages 45
8 Rational, real and complex numbers 49
9 Polynomials 55
10 Real polynomials, complex polynomials 65
11 Factoring polynomials 77
12 Fields 85
13 Field extensions 95
14 Formal power series 105
15 Resultants and discriminants 117
16 Permuting roots 131
17 Morphisms 145
18 Galois theory 155
19 Plane algebraic curves 165
20 Where plane algebraic curves intersect 183
21 Quotient rings 193
22 Field extensions and algebraic curves 199
23 The projective plane 207
24 Algebraic curves in the projective plane 221
25 Counting intersections of curves 229
26 Families of plane algebraic curves 239
27 Elliptic curves 251
28 The tangent line 255
29 Inflection points 263
30 Conics and quadratic forms 273
31 Projective duality 279
32 Polynomial equations have solutions 283
vii
viii Contents
33 Blow up 289
34 Schubert calculus 295
35 More projective planes 303
Hints 323
Bibliography 369
List of notation 371
Index 373
Chapter 1
The integers
God made the integers; all else is the work of man.
— Leopold Kronecker
Notation
We will write numbers using notation like 1234567.12345, using a decimal point .
at the last integer digit, and using thin spaces to separate out every 3 digits before
or after the decimal point. You might prefer 1,234,567·123,45 or 1,234,567.123,45,
whicharealsofine. Wereservethe·symbolformultiplication,writing2·3=6rather
than 2×3=6.
The laws of integer arithmetic
We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.
Through the unknown, remembered gate
When the last of earth left to discover
Is that which was the beginning;
At the source of the longest river
The voice of the hidden waterfall
And the children in the apple-tree
Not known, because not looked for
But heard, half-heard, in the stillness
Between two waves of the sea.
— T.S. Eliot
Little Gidding, Four Quartets
The integers are the numbers ...,−2,−1,0,1,2,.... Let us distill their essential
properties, using only the concepts of addition and multiplication.
Addition laws:
a. The associative law: For any integers a,b,c: (a+b)+c=a+(b+c).
1
2 The integers
b. Theidentitylaw:Thereisaninteger0sothatforanyintegera: a+0=a.
c. Theexistenceofnegatives:foranyintegera: thereisanintegerb(denoted
by the symbol −a) so that a+b=0.
d. The commutative law: For any integers a,b: a+b=b+a.
Multiplication laws:
a. The associative law: For any integers a,b,c: (ab)c=a(bc).
b. The identity law: There is an integer 1 so that for any integer a: a1=a.
c. The zero divisors law: For any integers a,b: if ab=0 then a=0 or b=0.
d. The commutative law: For any integers a,b: ab=ba.
The distributive law:
a. For any integers a,b,c: a(b+c)=ab+ac.
Sign laws:
Certain integers are called positive.
a. The succession law: An integer b is positive just when either b = 1 or
b=c+1 for a positive integer c.
b. Determinacy of sign: Every integer a has precisely one of the following
properties: a is positive, a=0, or −a is positive.
We write a<b to mean that there is a positive integer c so that b=a+c.
The law of well ordering:
a. Any nonempty collection of positive integers has a least element; that is
to say, an element a so that every element b satisfies a<b or a=b.
All of the other arithmetic laws we are familiar with can be derived from these. For
example,theassociativelawforaddition,appliedtwice,showsthat(a+b)+(c+d)=
a+(b+(c+d)), and so on, so that we can add up any finite sequence of integers, in
any order, and get the same result, which we write in this case as a+b+c+d. A
similar story holds for multiplication.
Ofcourse,wewrite1+1as2,and1+1+1as3andsoon. Writea>btomean
b<a. Write a≤b to mean a<b or a=b. Write a≥b to mean b≤a. Write |a| to
mean a, if a≥0, and to mean −a otherwise, and call it the absolute value of a. An
integer a is negative if −a is positive.