Appendix

APP., §1. THE NATURAL NUMBERS

The purpose of this appendix is to show how the integers can be ob­tained axiomatically using only the terminology and elementary proper­ties of sets. The rules of the game from now on allow us to use only sets and mappings.

We assume given once for all a set N called the set of natural numbers, and a map <T: N -+ N, satisfying the following (Peano) axioms:

NN 1. There is an element 0 E N.

NN 2. We have <T(O) =1= 0 and if we let N + denote the subset of N con­sisting of all n E N, n =1= 0, then the map x 1--4 <T(x) is a bijection between Nand N +.

NN 3. If S is a subset of N, if OES, and if <T(n) lies in S whenever n lies in S, then S = N.

We often denote <T(n) by n' and think of n' as the successor of n. The reader will recognize NN 3 as induction.

We denote <T(O) by l. Our next task is to define addition between natural numbers.

Lemma 1.1 Let f: N -+ Nand g: N -+ N be maps such that

f(O) = g(O)

Then f = g.

and { fen') = fen)', g(n') = g(n)'.

246 APPENDIX [APP., §1]

Proof Let S be the subset of N consisting of all n such that

fen) = g(n).

Then S obviously satisfies the hypotheses of induction, so S = N, thereby proving the lemma.

For each mEN, we wish to define m + n with nEN such that

and m + n' = (m + n)' for all nEN.

By Lemma 1.1, this is possible in only one way. If m = 0, we define 0 + n = n for all n E N. Then (lm) is obviously sat­

isfied. Let T be the set of mEN for which one can define m + n for all n E N in such a way that (lm) is satisfied. Then 0 E T. Suppose mET. We define for all n E N,

m'+O=m' and m' + n = (m + n)'. Then

m' + n' = (m + n')' = «m + n)')' = (m' + n)'.

Hence (lm,) is satisfied, so m' E T. This proves that T = N, and thus we have defined addition for all pairs (m, n) of natural numbers.

The properties of addition are easily proved.

Commutativity. Let S be the set of all natural numbers m such that

m+n=n+m for all nEN.

Then 0 is obviously in S, and if mE S, then

m' + n = (m + n)' = (n + m)' = n + m',

thereby proving that S = N, as desired. Associativity. Let S be the set of natural numbers m such that

(m + n) + k = m + (n + k) for all n, kEN.

Then 0 is obviously in S. Suppose mE S. Then

(m' + n) + k = (m + n)' + k, = «m + n) + k)'

= (m + (n + k)' = m' + (n + k),

thereby proving that S = N, as desired.

Cancellation law. Let m be a natural number. We shall say that the cancellation law holds for m if for all k, n E N satisfying m + k = m + n we

[APP., §1] THE NATURAL NUMBERS 247

must have k = n. Let S be the set of m for which the cancellation law holds. Then obviously 0 E S, and if mE S, then

m'+k=m'+n implies (m + k)' = (m + n)'.

Since the mapping x ~ x' is injective, it follows that m + k = m + n, whence k = n. By induction, S = N.

For multiplication, and other applications, we need to generalize Lemma 1.1.

Lemma 1.2. Let S be a set, and q>: S ~ S a map of S into itself. Let f, g be maps of N into S. If

f(O) = g(O) and {f(n') = q> 0 f(n), g(n') = q> 0 g(n)

for all nEN, then f = g.

Proof Trivial by induction.

For each natural number m, it follows from Lemma 1.2 that there is at most one way of defining a product mn satisfying

mO= 0 and mn' = mn + m for all nEN.

We in fact define the product this way in the same inductive manner that we did for addition, and then prove in a similar way that this product is commutative, associative, and distributive, that is

m(n + k) = mn + mk

for all m, n, kEN. We leave the details to the reader. In this way, we obtain all the properties of a ring, except that N is

not an additive group: We lack additive inverses. Note that 1 is a unit element for the multiplication, that is 1m = m for all mEN.

It is also easy to prove the multiplicative cancellation law, namely if mk = mn and m #- 0, then k = n. We also leave this to the reader. In particular, if mn #- 0, then m #- 0 al1d n #- O.

We recall that an ordering in a set X is a relation x ~ y between cer­tain pairs (x, y) of elements of X, satisfying the conditions (for all x, y, ZEX):

PO 1. We have x ~ x.

PO 2. If x ~ y and y ~ Z, then x ~ z.

PO 3. If x ~ y and y ~ x, then x = y.

248 APPENDIX [APP., §1]

The ordering is called a total ordering if given x, y E X we have x ~ y or y ~ x. We write x < y if x ~ y and x ¥- y.

We can define an drdering in N by defining n ~ 11i if there exists kEN such that m = n + k. The proof that this is an ordering is routine and left to the reader. This is in fact a total ordering, and we give the proof for that. Given a natural number m, let Cm be the set of n E N such that n ~ m or m ~ n. Then certainly 0 E Cm • Suppose that n E Cm • If n = m, then n' = m + 1, so m ~ n'. If n < m, then m = n + k' for some kEN, so that

m = n + k' = (n + k)' = n' + k,

and n' ~ m. If m ~ n, then for some k, we have n = m + k, so that n + 1 = m + k + 1 and m ~ n + 1. By induction, Cm = N, thereby show­ing our ordering is total.

It is then easy to prove standard statements concerning inequalities, e.g.

m < n if and only if m + k < n + k for some kEN,

m < n if and only if mk < nk for some kEN, k ¥- o.

One can also replace "for some" by "for all" in these two assertions. The proofs are left to the reader. It is also easy to prove that if m, n are natural numbers and m ~ n ~ m + 1, then m = n or n = m + 1. We leave the proof to the reader.

We now prove the first property of integers mentioned in Chapter I, §2, namely the well-ordering:

Every non-empty subset S of N has a least element.

To see this, let T be the subset of N consisting of all n such that n ~ x for all XES. Then 0 E T, and T ¥- N. Hence there exists mET such that m + 1 rt T (by induction!). Then mES (otherwise m < x for all XES which is impossible). It is then clear that m is the smallest element of S, as desired.

In Chapter IX, we assumed known the properties of finite cardinali­ties. We shall prove these here. For each natural number n¥-O let I n be the set of natural numbers x such that 1 ~ x ~ n.

If n = 1, then I n = {1}, and there is only a single map of J 1 into itself. This map is obviously bijective. We recall that sets A, B are said to

have the same cardinality if there is a bijection of A onto B. Since a composite of bijections is a bijection, it follows that if

card(A) = card(B)

then card(A) = card(C).

and card(B) = card(C),

[APP., §1] THE NATURAL NUMBERS 249

Let m be a natural number ~ 1 and let kEJm,. Then there is a bijec­tion between

and

defined in the obvious way: We let f: J m, - {k} -+Jm be such that

f:xl-+x if x < k,

f: x 1-+ a-lex) if x> k.

We let g: Jm -+ Jm, - {k} be such that

g: x 1-+ x if x < k,

g: x 1-+ a(x) if x~k.

Then fog and go f are the respective identities, so f, g are bijections.

We conclude that for all natural numbers m ~ 1, if

is an injection, then h is a bijection.

Indeed, this is true for m = 1, and by induction, suppose the statement true for some m ~ 1. Let

be an injection. Let r E J m' and let s = <p(r). Then we can define a map

<Po: J m, - {r} -+ J m, - {s}

by x 1-+ <p(x). The cardinality of each set Jm, - {r} and Jm, - {s} is the same as the cardinality of J m. By induction, it follows that <Po is a bijec­tion, whence <p is a bijection, as desired.

We conclude that if 1 ~ m < n, then a map

cannot be injective.

For otherwise by what we have seen,

and hence

fen) = f(x)

for some x such that 1 ~ x ~ m, so f is not injective.

250 APPENDIX [APP., §2]

Given a set A, we shall say that card(A) = n (or the cardinality of A is n, or A has n elements) for a natural number n ~ 1, if there is a bijection of A with I n • By the above results, it follows that such a natural number n is uniquely determined by A. We also say that A has cardi­nality 0 if A is empty. We say that A is finite if A has cardinality n for some natural number n. It is then an exercise to prove the following statements:

If A, B are finite sets, and A n B is empty, then

card(A) + card(B) = card(A u B).

Furthermore,

card(A) card(B) = card(A x B).

We leave the proofs to the reader.

APP., §2. THE INTEGERS

Having the natural numbers, we wish to define the integers. We do this the way it is done in elementary school; there is no better way.

For each natural number n"# 0 we select a new symbol denoted by - n, and we denote by Z the set consisting of the union of N and all the symbols -n for nEN, n "# O. We must define addition in Z. If x, YEN we use the same addition as before. For all x E Z, we define

0+ x = x + 0 = x.

This is compatible with the addition defined in §1 when XEN. Let m, nEN and neither n nor m = O. If m = n + k with kEN we

define:

(a) m + (-n) = (-n) + m = k. (b) (- m) + n = n + (-m) = - k if k "# 0, and = 0 if k = O. (c) (-m)+(-n)= -(m+n).

Given x, Y E Z, if not both x, yare natural numbers, then at least one of the situations (a), (b), (c) applies to their addition.

It is then tedious but routine to verify that Z is an additive group. Next we define multiplication in Z. If x, YEN we use the same

multiplication as before. For all x E Z we define Ox = xO = O. Let m, nEN and neither n nor m = O. We define:

(-m)n = n( -m) = -(mn) and (-m)( -n) = mn.

[APP., §3] INFINITE SETS 251

Then it is routinely verified that Z is a commutative ring, and is in fact entire, its unit element being the element 1 in N. In this way we get the integers.

Observe that Z is an ordered ring in the sense of Chapter IX, §1 be­cause the set of natural numbers n :f= 0 satisfies all the conditions given in that chapter, as one sees directly from our definitions of multiplication and addition.

APP., §3. INFINITE SETS

A set A is said to be infinite if it is not finite (and in particular, not empty).

We shall prove that an infinite set A contains a denumerable subset. For each nonempty subset T of A, let X T be a chosen element of T. We prove by induction that for each positive integer n we can find uniquely determined elements xl, ... ,XnEA such that Xl = x A is the chosen element corresponding to the set A itself, and for each k = 1, ... ,n - 1, the ele­ment X k + l is the chosen element in the complement of {xl, ... ,Xk}. When n = 1, this is obvious. Assume the statement proved for n> 1. Then we let Xn + 1 be the chosen element in the complement of {Xl'''' ,xn}. If Xl'''' ,Xn are already uniquely determined, so is X n+ l' This proves what we wanted. In particular, since the elements Xl'''' ,Xn are distinct for all n, it follows that the subset of A consisting of all elements Xn is a denumerable subset, as desired.

Index

A

Abelian extension 174, 178 Abelian group 14, 49 Absolute value 201, 214, 219 Additive group 14 Adic expansion 11, 92 Adjoin 61 Affine group or map 144, 145 Algebraic 158 Algebraic extension 181 Algebraic integer 195 Algebraically closed 78, 136, 161, 181,

220 Algebraically independent 111 Alternating group 41 Arbitrarily large 203 Archimedean 204, 212 Associated linear map 126 Associativity 13 Automorphism of group 27 Automorphism or ring 61, 71, 169 Automorphism of vector space 122 Axiom of choice 163

B

Basis 115, 134, 139, 141, 147, 229 Bernstein's theorem 233 Bijective 20 Binomial coefficients 5 Borel subgroup 153

C

Canonical homomorphism 33, 138 Cardinality 231 Cauchy sequence 198, 203 Center 35, 148, 150 Characteristic 71 Chinese Remainder Theorem 12, 65 Classs field theory 195 Coefficients of polynomial 75, 99 Common divisor 5 Commutative 14, 55 Commutative group 35, 153 Commutator subgroup 35 Complement 225 Complete 204, 212 Completion 198, 212 Complex numbers 218 Component 123 Composite field 175 Composite mapping 22 Congruence 10, 62 Conjugate 35 Constant term 75 Converge 199, 204 Coordinate 115 Coordinate vector 115 Coset 30 Covering 234 Cubic extension 176 Cycle 41 Cyclic group 45 Cyclotomic equation 192

254

D

Decimal expansion 215 Degree of element 159 Degree of extension 159 Degree of polynomial 75 Denominator 9 Denumerable 218, 231 Derivation 67 Derivative 83 Dimension 121 Direct product 15 Direct sum 47 Discriminant 176 Disjoint sets 225, 234 Distributive 57 Divide 5, 80, 102 Division ring 133 Divisor of zero 57

E

Eigenvalue 145 Eigenvector 145 Eisenstein criterion 96, 107 Element 1 Embedding 69, 163 Empty set 1 Endomorphism 54, 119, 128, 129 Entire ring 57~·

Equivalence class 10 Equivalence relation 10 Euclidean algorithm 4, 76 Euler phi function 12 Evaluation 109 Even permutation 40 Exact 135 Exponent 45, 49 Extension of embedding 164 Extension of fields 158

F

Factor group 33 Factor module 137 Factor ring 63 Factorial ring 102 Factorization 83 Family 223 Fan 147 Field 57 Finite abelian group 49 Finite extensio~ 158 Finite field 184

INDEX

Finite group 15 Finite order 46 Finite period 46, 145 Finitely generated 142 Fixed field 168 Formal polynomial 99 Free 134, 198 Free abelian group 139 Frobenius automorphism 188, 195

G

Galois extension 171 Galois group 172 Gauss' lemma 95, 105 General linear group 122, 144, 147 Generate 6, 17, 114, 128 Generators 17, 45, 59, 79, 114, 128 GL 122, 147 Greatest common divisor 5, 80, 102 Greatest element 227 Group 13

H

Homomorphism of groups 24 Homomorphism of modules or vector

spaces 117 Homomorphism of rings 61

I

Ideal 6, 58 Identity 21, 124 Image 19 Independent variables 111 Index 30 Indexed 224 Induced ordering 199, 227 Induction 2 Inductively ordered 228 Infinite order 45 Infinite period 45 Infinite set 251 Injective 20 Inner automorphism 29, 34 Integers 250 Intersection 1 Invariant 136 Inverse 14 Inverse image 32, 225 Inverse map 21, 118 Invertible 144 Irreducible 81, 102, 159 Isomorphism 26, 61, 119, 129, 163

K

Kernel 26, 118, 129

L

Leading coefficient 75 Least common multiple 11 Least upper bound 204, 227 Left coset 30 Left ideal 57 Limit 204 Linear map 117, 125 Linear polynomial 76 Linearly dependent 114, 120 Linearly independent 114, 229

M

Mapping 19 Matrix 122 Maximal element 227 Maximal ideal 64, 66 Maximal subset of linearly

independent elements 116 Mobius function 66 Module 127 Modulo 10, 64, 137 Monic 83 Monomial 99 Multiple root 84 Multiplicative function 66 Multiplicative group 13 Multiplicity of root 83

N

Natural number 246 Negative 199 Nilpotent 49, 146 Norm 167 Normal extension 169 Normal subgroup 32 Normalizer 35 Null sequence 198, 207 Numerator 9

o Odd permutation 40 Onto 20 Operation 156 ' Orbit 157 Order of linear group 147

INDEX

Order preserving 200 Ordering of ring 199, 247 Ordering of sets 226 Over 164

P

p-adic 214 Partial fraction 88 Partial ordering 226 Period 45 Permutation 20, 23, 34 PGL 150 p-group 48 Polynomial 73, 99, 108 Polynomial function 72, 100 Polynomial ring 94 Positive 199 Prime 7, 101, 102 Prime field 185 Prime number 7 Primitive root 191 Primitive root of unity 174 Principal ideal 59 Principal ring 101 Product 15

255

Projective special linear group 151 Proper 1 PSL 151

Q Quadratic 175 Quadratic residue 191 Quaternion group 18 Quotient field 69

R

Radicals 179 Rational function 69 Rational number 9 Real number 209 Relation 246 Relatively prime 7, 81, 104 Remainder 15 Representation 196 Representative 10, 62 Restriction 20, 61, 164 Right coset 30 Right ideal 59 Right multiplication 130 Ring 54 Root 74 Root of unity 47, 174, 192

256

s Scalar matrix 124 Schroeder-Bernstein theorem 233 Schur's lemma 133 Sequence 203, 223 Set 1 Shafarevich conjecture 198 Sign 39 Simple module 133 Solvable extension 179 Solvable group 42, 179 Special projective linear group 151 Splitting field 169 Square matrix 123 Strictly inductively ordered 228 Subgroup 16 Submodule 127, 137 Subring 57 Subset 1 Subspace 113 Substitution 98 Sufficiently large 203 Sum 28, 119 Surjective 20 Symmetric group 43, 180

T

Torsion group 48 Totally ordered 247 Trace 167 Transcendental 98

INDEX

Translation 27, 145 Transposition 34 Trivial 15 Twin primes 10 Two-sided ideal 49, 131 Type of a group 49

U

Union 1,224 Unique factorization 7, 31, 102 Unit 47, 48, 145 Unit element 14, 55 Unit ideal 6, 58 Unit matrix 124 Upper bound 204, 227

v Value 19 Variable 99 Vector space 112

w Wedderburn-Rieffel theorem 132 Well-ordering 2

Z

Zero element 14 Zorn's lemma 228

Undergraduate Texts in Mathematics

Martin: The Foundations of Geometry and the Non-Euclidean Plane.

Martin: Transformation Geometry: An Introduction to Symmetry.

MillmanlParker: Geometry: A Metric Approach with Models.

Owen: A First Course in the Mathematical Foundations of Thermodynamics.

Prenowitz/Jantosciak: Join Geometrics.

Priestly: Calculus: An Historical Approach.

Protter/Morrey: A First Course in Real Analysis.

Protter/Morrey: Intermediate Calculus.

Ross: Elementary Analysis: The Theory of Calculus.

continued from ii

ScharlauiOpolka: From Fermat to Minkowski.

Sigler: Algebra.

Simmonds: A Brief on Tensor Analysis.

SingerlThorpe: Lecture Notes on Elementary Topology and Geometry.

Smith: Linear Algebra. Second edition.

Smith: Primer of Modem Analysis.

Thorpe: Elementary Topics in Differential Geometry.

Troutman: Variational Calculus with Elementary Convexity.

Wilson: Much Ado About Calculus.

Springer Books on Elementary Mathematics by Serge Lang

The Beauty of Doing Mathematics 1985, ISBN 96149-6

Calculus of Several Variables 1987, ISBN 96405-3

Complex Analysis 1985, ISBN 96085-6

A First Course in Calculus 1986, ISBN 96201-8

Geometry. A High School Course (with G. Murrow) 1983, ISBN 90727-0

Introduction to Linear Algebra 1986, ISBN 96205-0

Linear Algebra 1987, ISBN 96412-6

MA TH! Encounters with High School Students 1985, ISBN 96129-1

Undergraduate Algebra 1987, ISBN 96404-5

Undergraduate Analysis 1983, ISBN 90800-5