Springer number theory b (springer)

NUMBER THEORY
An Introduction to Mathematics: Part B


NUMBER THEORY
An Introduction to Mathematics: Part B

BY
WILLIAM A. COPPEL

Q
- Springer


Library of Congress Control Number: 2005934653
PARTA

ISBN-10:0-387-29851-7

e-ISBN: 0-387-29852-5

ISBN-13: 978-0387-29851-1

PART B

ISBN-10:0-387-29853-3

e-ISBN: 0-387-29854-1

ISBN-13: 978-0387-29853-5

2-VOLUME SET
ISBN-10: 0-387-30019-8

e-ISBN:0-387-30529-7

ISBN-13: 978-0387-30019-1

Printed on acid-free paper

AMS Subiect Classifications: 1 1-xx. 05820.33E05
O 2006 Springer Science+Business Media, Inc

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Printed in the United States of America


Contents
Part A
Preface

I

The expanding universe of numbers

Sets, relations and mappings
Natural numbers
Integers and rational numbers
Real numbers
Metric spaces
Complex numbers
Quaternions and octonions
Groups
Rings and fields
Vector spaces and associative algebras
Inner product spaces
Further remarks
12 Selected references

I1

Divisibility
1
2
3
4
5
6
7
8

Greatest common divisors
The Bezout identity
Polynomials
Euclidean domains

Congruences
Sums of squares
Further remarks
Selected references


Contents

I11

More on divisibility
1
2
3
4
5
6

IV

The law of quadratic reciprocity
Quadratic fields
Multiplicative functions
Linear Diophantine equations
Further remarks
Selected references

Continued fractions and their uses
1 The continued fraction algorithm
2

3
4
5
6

7
8
9
V

Diophantine approximation
Periodic continued fractions
Quadratic Diophantine equations
The modular group
Non-Euclidean geometry
Complements
Further remarks
Selected references

Hadamard's determinant problem
1
2
3
4
5

6
7
8

9

What is a determinant?
Hadamard matrices
The art of weighing
Some matrix theory
Application to Hadamard's determinant problem
Designs
Groups and codes
Further remarks
Selected references


Contents

VI

Hensel's p-adic numbers
Valued fields
Equivalence
Completions
Non-archimedean valued fields
Hensel's lemma
Locally compact valued fields
Further remarks
Selected references
Notations
Axioms
Index
Part B

VII

The arithmetic of quadratic forms
1 Quadratic spaces
2 The Hilbert symbol
3 The Hasse-Minkowski theorem
4 Supplements
5 Further remarks
6 Selected references

VIII The geometry of numbers
1
2
3
4
5
6
7
8

Minkowski's lattice point theorem
Lattices
Proof of the lattice-point theorem, and some generalizations
Voronoi cells
Densest packings
Mahler's compactness theorem
Further remarks
Selected references

vii


...

Contents

Vlll

IX

The number of prime numbers
1
2
3
4

5
6
7
8
9
X

A character study
1
2
3
4
5

6
7
8
9
10

XI

Finding the problem
Chebyshev's functions
Proof of the prime number theorem
The Riemann hypothesis
Generalizations and analogues
Alternative formulations
Some further problems
Further remarks
Selected references

Primes in arithmetic progressions
Characters of finite abelian groups
Proof of the prime number theorem for arithmetic progressions
Representations of arbitrary finite groups
Characters of arbitrary finite groups
Induced representations and examples
Applications
Generalizations
Further remarks
Selected references

Uniform distribution and ergodic theory
1 Uniform distribution

2 Discrepancy
3 Birkhoff's ergodic theorem
4
5
6
7

Applications
Recurrence
Further remarks
Selected references


XI1

Elliptic functions
1 Elliptic integrals
2 The arithmetic-geometric mean

3
4
5
6
7
8

Elliptic functions

Theta functions
Jacobian elliptic functions
The modular function
Further remarks
Selected references

XI11 Connections with number theory
Sums of squares
Partitions
Cubic curves
Mordell's theorem
Further results and conjectures
Some applications
Further remarks
Selected references
Notations
Axioms
Index


VII

The arithmetic of quadratic forms

We have already determined the integers which can be represented as a sum of two squares.
Similarly, one may ask which integers can be represented in the form x2 + 2y2 or, more
generally, in the form ax2 + 2bxy + cy2, where a,b,c are given integers. The arithmetic theory
of binary quadratic forms, which had its origins in the work of Fermat, was extensively
developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to
quadratic forms in more than two variables, which was begun by them and is exemplified by

Lagrange's theorem that every positive integer is a sum of four squares, was continued during
the 19th century by Dirichlet, Hermite, H.J.S. Smith, Minkowski and others. In the 20th
century Hasse and Siege1 made notable contributions. With Hasse's work especially it became
apparent that the theory is more perspicuous if one allows the variables to be rational numbers,
rather than integers. This opened the way to the study of quadratic forms over arbitrary fields,
with pioneering contributions by Witt (1937) and Pfister (1965-67).
From this vast theory we focus attention on one central result, the Hasse-Minkowski

theorem. However, we first study quadratic forms over an arbitrary field in the geometric
formulation of Witt. Then, following an interesting approach due to Frohlich (1967), we study
quadratic forms over a Hilbertfield.

1 Quadratic spaces
The theory of quadratic spaces is simply another name for the theory of quadratic forms.
The advantage of the change in terminology lies in its appeal to geometric intuition. It has in
fact led to new results even at quite an elementary level. The new approach had its debut in a
paper by Witt (1937) on the arithmetic theory of quadratic forms, but it is appropriate also if one
is interested in quadratic forms over the real field or any other field.
For the remainder of this chapter we will restrict attention to fields for which 1 + 1 # 0.
Thus the phrase 'an arbitrary field' will mean 'an arbitrary field of characteristic # 2'. The


342

VII. The arithmetic of quadratic forms

proofs of many results make essential use of this restriction on the characteristic. For any field
F , we will denote by F X the multiplicative group of all nonzero elements of F. The squares in
FX form a subgroup Fx2 and any coset of this subgroup is called a square class.
Let V be a finite-dimensional vector space over such a field F. We say that V is a

quadratic space if with each ordered pair u,v of elements of V there is associated an element
(u,v) of F such that
(i) (ul + u2,v) = (ul,v) + (u2,v) for all ul7u2,v E V ;
(ii) (au,v) = a(u,v) for every a E F and all u,v E V ;
(iii) (u,v) = (v,u) for all u,v E V.
It follows that
(i)' (u,vl + v2) = (u,vI) + (u,v2) for all u,vl,v2 E V;
(ii)' (u,av) = a(u,v) for every a E F and all u,v E V.

Let e l , ...,en be a basis for the vector space V. Then any u,v
expressed in the form
u=

C ;,1 cjq, V = C

E

V can be uniquely

"rlej,

is a quadratic form with coefficients in F. The quadratic space is completely determined by the
quadratic form, since
(u,v) = {(u + 1.?,U+ v) - (u,u) - (v,v))/2.
(1)
Conversely, for a given basis el ,...,en of V, any nxn symmetric matrix A = (ajk) with
elements from F , or the associated quadratic form f(x) = xlAx, may be used in this way to give
V the structure of a quadvatic space.
Let el', ...,en' be any other basis for V. Then

where T = ( ( z i j ) is an invertible 11x1~matrix with elements from F. Conversely, any such matrix

T defines in this way a new basis el ',...,en1. Since


1 . Quadratic spaces

where

pjh = (ejf,eh?,it follows that
A = TtBT.

Two symmetric matrices A,B with elements from F are said to be congruent if (2) holds for
some invertible matrix T with elements from F. Thus congruence of symmetric matrices
con-esponds to a change of basis in the quakatic space. Evidently congmence is an equivalence
relation, i.e. it is reflexive, symmetric and transitive. Two quadratic forms are said to be
equivalent over F if their coefficient matrices are congruent. Equivalence over F of the
quadratic foimsf and g will be denoted byf -F g or simplyf - g.
It follows from (2) that
det A

=

(det

n2det B.

Thus, although det A is not uniquely determined by the quadratic space, if it is nonzero, its
square class is uniquely dete~mined.By abuse of language, we will call any representative of
this square class the determinant of the quadratic space V and denote it by det V.

Although quadratic spaces are better adapted for proving theorems, quadratic forms and
symmetric matrices are useful for computational purposes. Thus a familiarity with both
languages is desirable. However, we do not feel obliged to give two versions of each definition
or result, and a version in one language may later be used in the other without explicit comment.

A vector v is said to be orthogonal to a vector u if (u,v) = 0. Then also u is orthogonal to
v. The o~~thogonal
conzplement U' of a subspace U of V is defined to be the set of all v E V
such that (u,v) = 0 for every u E U. Evidently U' is again a subspace. A subspace U will be
said to be non-singular if U n u -=~{O}.
The whole space V is itself non-singulas if and only if V' = (0). Thus V is non-singular if
and only if some, and hence every, sy~nmetricmatrix describing it is non-singulau, i s . if and
only if det V # 0.
We say that a quadratic space V is the orthogonal sum of two subspaces Vl and V . ,and we
write V = V1 IV2, if V = VI + V2,V1 n V2 = ( 0 ) and (v1,v2)= 0 for all vl E V1, v2 E V2.
If Al is a coefficient matrix for V1 and A2 a coefficient matrix for V2, then

is a coefficient matrix for V = V1 IV2. Thus det V = (det Vl)(det V2). Evidently V is nonsingulau if and only if both V1 and V2 are non-singulas.


VII. The arithmetic of quudratic forms

If W is any subspace supplementasy to the orthogonal complement V' of the whole space
V, then V = v - I
~ Wand W is non-singulas. Many problems for arbitrary quadsatic spaces may
be reduced in this way to non-singular quadratic spaces.

PROPOSITION
1 If a quadratic space V co~ltailzsa vector u such that (u,u) # 0 , then

where U = <u> is the one-dimensional subspace sparzned by u.
Proof For any vector v E V, put v' = v - au, where a = (v,zc)/(u,u). Then (vf,u)= 0 and
hence v' E u'. Since U n U' = { 0 ) ,the result follows. 17
A vector space basis u l ,...,u, of a quadratic space V is said to be an orthogonal basis if
(uj,uk)= 0 whenever j # k.
PROPOSITION 2 Any quadintic space V has an ortlzogonal basis.

Proof If V has dimension 1, there is nothing to prove. Suppose V has dimension n > 1 and the
result holds for quadsatic spaces of lower dimension. If (v,v) = 0 for all v E V, then any basis
is an orthogonal basis, by (1). Hence we may assume that V contains a vector ul such that
(u,,ul) # 0. If U 1is the 1-dimensional subspace spanned by ul then, by Psoposition 1,

By the induction hypothesis

has an orthogonal basis u2,...,u,, and u1,u2,...,un is then an

orthogonal basis for V.
Proposition 2 says that any symnetic matrix A is congruent to a diagonal matsix, or that
the cossesponding quadratic form f is equivalent over F to a diagonal form F1c12 + ... + 8,5,2.
Evidently det f = 6,...8, and f is non-singular if and only if 6j # 0 (1 I j I n). If A # 0 then,
by Propositions 1 and 2, we can take F1 to be any element of FXwhich is represented by$
Here y E FXis said to be wpreseizted by a quadratic space V over the field F if there
exists a vector v E V such that (v,v) = y.
As an application of Psoposition 2 we prove

PROPOSITION3 If U is a non-singular subspace of the quadratic space V, then

v = U I uL.
Proof Let u , ,...,u, be an orthogonal basis for U . Then (uJ,u,) # 0 ( 1 I j I m), since U is
non-singular. For any vector v E V, let u = allcl + ... + a,,~,,, where aJ = (v,uj)/(uJ,uj)

for


345

I . Quadratic spaces

each j. Then u E U and (u,uj) = (v,uj) ( 1 I j I m). Hence v - u
the result follows.

E

u'.

since u n U' = { 0 ) ,

It may be noted that if U is a non-singular subspace and V = U IW for some subspace W ,
then necessarily W = U-L. For it is obvious that W c U' and dim W = dim V - dim U = dim u',
by Proposition 3.
PROPOSITION 4 Let V be a non-singular quadratic space. If v l ,...,v , are linearly
independent vectors in V then, for arbitrary q l ,...,qm E F , there exists a vector v E V such

that (vj,v)= qj (1 I j I m ) .
Moreover, if U is any subspace of V, then
(i) d i m ~ + d i r n ~ ' = d i m ~ / ;
(ii) UU = U;

(iii)

u -is~non-singular ifand only if U is non-singular.

Proof There exist vectors v,+~ ,...,v, E V such that v l ,...,v, form a basis for V. If we put
ajk= (vj,vk)then, since V is non-singular, the nxn symmetric matrix A = (ajk)is non-singular.
Hence, for any q 1,...,q n E F , there exist unique k1,...,5, E F such that v = c l v l + ... + cnv,
satisfies
(

~

1

3

= ) ql?... (vn7v)= rln.

~

This proves the first part of the proposition.
By taking U = <vl ,...,v,> and q = ... = q , = 0, we see that dim u -=~n - m. Replacing

U by u-',we obtain dim ULL= dim U. Since it is obvious that U c UU, this implies U = UU.
Since U non-singular means U n U' = { 0 ) ,(iii) follows at once from (ii).
We now introduce some further definitions. A vector u is said to be isotropic if u # 0 and

(u,u) = 0. A subspace U of V is said to be isotropic if it contains an isotropic vector and
anisotropic otherwise. A subspace U of V is said to be totally isotropic if every nonzero
vector in U is isotropic, i.e. if U L u'. According to these definitions, the trivial subspace 10)
is both anisotropic and totally isotropic.
A quadratic space V over a field F is said to be universal if it represents every y E FX,i.e.
if for each y E FXthere is a vector v E V such that (v,v) = y.

PROPOSITION
5 If a non-singular quadratic space V is isotropic, then it is universal.
Proof Since V is isotropic, it contains a vector u # 0 such that (u,u) = 0. Since V is nonsingular, it contains a vector w such that (u,w) # 0. Then w is linearly independent of u and by


346

VII. The arithmetic of quadratic forms

replacing w by a scalar multiple we may assume (u,w) = 1. If v = a u
a = { y - (w,w)}/2.

+ w, then (v,v) = y for

On the other hand, a non-singular universal quadratic space need not be isotropic. As an
example, take F to be the finite field with three elements and V the 2-dimensional quadratic
space corresponding to the quadratic form k 1 2 + C22.

PROPOSITION
6 A non-singular quadratic form f(51,...,5n) with coefficientsfrom a field
F represents y E F X if and only if the quadratic f o m

is isotropic.
Obviously if f ( x l , ...,x,) = y and xo = 1, then g(xo,xl,...,x,) = 0. Suppose on the
other hand that g(xo,xl,...,x,) = 0 for some xj E F, not all zero. If xo # 0, then f certainly
represents y. If xo = 0, then f is isotropic and hence, by Proposition 5, it still represents y. 0

Proof

PROPOSITION
7 Let V be a non-singular isotropic quadratic space. If V = U IW , then
there exists y E F X such that, for some u E U and w E W ,

Proof Since V is non-singular, so also are U and W, and since V contains an isotropic vector
v', there exist U ' E U , W' E W , not both zero, such that

If this common value is nonzero, we are finished. Otherwise either U or W is isotropic.
Without loss of generality, suppose U is isotropic. Since W is non-singular, it contains a vector
w such that (w,w) # 0, and U contains a vector u such that (u,u) = - (w,w), by Proposition 5.

0
We now show that the totally isotropic subspaces of a quadratic space are important for an
understanding of its structure, even though they are themselves trivial as quadratic spaces.
PROPOSITION 8 All maximal totally isotropic subspaces of a quadratic space have the

same dimension.
Proof Let Ul be a maximal totally isotropic subspace of the quadratic space V. Then U 1c u~'
and ull\ U1contains no isotropic vector. Since V' c u
~', it follows that V' r U1. If V' is a


I . Quadratic spaces

347

subspace of V supplementary to v', then V' is non-singular and U 1 = V' + U 1 ' , where
U1'c V'. Since Ul' is a maximal totally isotropic subspace of V',this shows that it is sufficient
to establish the result when V itself is non-singular.
Let U 2 be another ~naximaltotally isotropic subspace of V . Put W = U 1 n U2 and let

W1,W2be subspaces supplementary to Win U1,U2respectively. We are going to show that
w2 n w,' = 10).
Let v

E

W 2 n wli. Since W 2 c U2,v is isotropic and v

E

u2' c w'.

Hence v

E

ul'

and actually v E U 1 ,since v is isotropic. Since W 2 c U 2 this implies v E W, and since
W n W2 = { 0 ) this implies v = 0.
It follows that dim W 2 + dim wI1 5 dim V . But, since V is now assumed non-singular,
dim W 1 and, for the same
dim W1 = dim V - dim w I 1 , by Proposition 4. Hence dim W 2 I
reason, dim W 1I dim W2. Thus dim W2 = dim W 1 ,and hence dim U2 = dim U1.
We define the index, ind V , of a quadsatic space V to be the dimension of any maximal
totally isotsopic subspace. Thus V is anisobopic if and only if ind V = 0.
A field F is said to be ordered if it contains a subset P of positive elements, which is
closed under addition and multiplication, such that F is the disjoint union of the sets {0}, P and
- P = {- x: x E P}. The rational field Q and the real field IW are ordered fields, with the usual
interpretation of 'positive'. For quadsatic spaces over an ordered field there are other useful

notions of index.

A subspace U of a quadratic space V over an ordered field F is said to be positive definite
if (u,u) > O for all nonzero u E U and ~zegativedefinite if (u,u) < O for all nonzero 14 E U .
Evidently positive definite and negative definite subspaces are anisotropic.

PROPOSITION
9 All maxinzal positive definite subspaces of a quadratic space V over an
ordered.field F have the s u m dimensio/~.
Proof Let U+be a maximal positive definite subspace of the quadratic space V. Since U+is
certainly non-singular, we have V = U+IW , where W = u+', and since U+ is maximal,
(w,w) I 0 for all w E W. Since U+c V , we have V' c W. If U-is a maximal negative definite
subspace of W , then in the same way W = U- IUo,where Uo = U-' n W . Evidently Uo is
totally isotropic and Uo c v'. In fact Uo = v'-, since U- n V' = {O). Since (v,v) 2 O for all
v E U+Iv', it follows that U- is a maximal negative definite subspace of V.
If U+'is another maximal positive definite subspace of V , then U+'n W = (0) and hence
dim U+'+ dim W = dim (U,'

+ W)

I dim V .


348

VII. The ar.itlmztic of yliudmtic fonns

Thus dim U+' I dim V - dim W = dim U+. But U+and U+'can be interchanged.
If V is a quadsatic space over an ordered field F, we define the positive index ind' V to be
the dimension of any maximal positive definite subspace. Similarly all maximal negative

definite subspaces have the same dimension, which we call the negative index of V and denote
by ind- V. The proof of Proposition 9 shows that
ind' V + ind- V + dim V' = dim V.

PROPOSITION
10 Let F denote the real field R or, more genemlly, an ordered field in
which every positive element is a squaw. The11 any nun-singular quaclvatic folm f in n
variables with coe8icients @om F is equivalerzt over F to a quadratic form

where p

E

{0,1, . . . , i l l is ~ m i q z d ydetermined by j: In fact,

Proof By Proposition 2, f is equivalent over F to a diagonal form 61q12+ ... + 6,q,2, where
6j z 0 (1 I j I n). We may choose the notation so that Fj > 0 for j I p and 6j < 0 for j > p .
('j > p ) now brings f to the form g.
The change of variables = 651/2q,('j S p ) , = (- !i)1/2qj
Since the corresponding quadratic space has a p-dimensional maximal positive definite
subspace, y = ind'f is uniquely determined. Similarly rz - p = ind- f, and the for~nulafor ind f
follows readily.

tj

cj

It follows that, for quadratic spaces over a field of the type considered in Proposition 10, a

subspace is anisokopic if and only if it is either positive definite or negative definite.

Proposition 10 completely solves the problem of equivalence for real quadsatic forms.
(The uniqueness of p is known as Sylvester's law of inertia.) It will now be shown that the
problem of equivalence for quadsatic forms over a finite field can also be completely solved.

LEMMA11

I f V is a rlorz-sirzgular 2-dit~~erzsiorzul
quadratic space over a firlite field E, of

(odd) ca~dinulityq, then V is u~zivelzral.
Proof By choosing an orthogonal basis for V we are reduced to showing that if a,P,y E IFqX,
then there exist t,q E [Fq such that at2+ Pq2 = y. As 5 runs through 5,, at2 takes (y + 1)/2
= 1 + ( q - 1)/2 distinct values. Similarly, as q runs through Fq, y - pq2 takes ( q + 1)/2
distinct values. Since ( q + 1)/2 + ( q + 1)/2 > q , there exist t,q E IF, for which a t 2 and
y - pq2 take the same value.


1 . Quadratic spaces

349

PROPOSITION12 Any non-singular quadratic form f in n variables over afinitefield IFq
is equivalent over [Fq to the quadratic form

where 6 = det f is the determinant o f f .
There are exactly two equivalence classes of non-singular quadratic forms in n
variables over F4, one consisting of those forms f whose determinant det f is a square in
LFqX ,and the other those for which det f is not a square in FqX.
Proof Since the first statement of the proposition is trivial for n = 1, we assume that n > 1 and
it holds for all smaller values of n. It follows from Lemma 11 that f represents 1 and hence, by

the remark after the proof of Proposition 2, f is equivalent over IFq to a quadratic form
E12 + g ( c 2 ,
Since f and g have the same determinant, the first statement of the

...,en).

proposition now follows from the induction hypothesis.
Since FqX contains (q - 1)/2 distinct squares, every element of

[FqX

is either a square or a

square times a fixed non-square. The second statement of the proposition now follows from the
first.
We now return to quadratic spaces over an arbitrary field. A 2-dimensional quadratic
space is said to be a hyperbolic plane if it is non-singular and isotropic.
PROPOSITION 13 For a 2-dimensional quadratic space V , the following statements are

equivalent:
(i) V is a hyperbolic plane;
(ii) V has a basis ul,u2 such that (ul,ul)= (u2,u2)= 0, (u1,u2)= 1;
(iii) V has a basis vl,v2 such that (vl,vl) = 1, (v2,v2)= - 1, (v1,v2)= 0;
(iv) - det V is a square in FX.

Proof Suppose first that V is a hyperbolic plane and let ul be any isotropic vector in V. If v is
any linearly independent vector, then (ul,v) # 0, since V is non-singular. By replacing v by a
scalar multiple we may assume (ul,v) = 1. If we put u2 = v + n u l , where a = - (v,v)/2,then

and ul,u2 is a basis for V.

If ul,u2 are isotropic vectors in V such that (u1,u2)= I, then the vectors vl = ul
and v2 = ul - u2/2 satisfy (iii), and if vl,v2 satisfy (iii) then det V = - 1.

+ u2/2


VII. The arithmetic of quadraticforms

Finally, if (iv) holds then V is certainly non-singular. Let wl,w2 be an orthogonal basis
= (wj,wj)(j = 1,2). By hypothesis, 8162= - y2, where y E FX. Since
for V and put

aj

ywl

+ 8 1 ~ is2 an isotropic vector, this proves that (iv) implies (i).

PROPOSITION14 Let V be a non-singular quadratic space. If U is a totally isotropic
subspace with basis ul,...,urn,then there exists a totally isotropic subspace U' with basis
u1',...,urnr such that
(uj,uk')= 1 or 0 according as j = k or j + k.
Hence U n U' = { 0 )and
U + U' = H1 I

... IH,,

where Hj is the hyperbolic plane with basis uj,ujl(1 I j I m).
Proof Suppose first that m = 1. Since V is non-singular, there exists a vector v E V such that
(ul,v) # 0. The subspace H1 spanned by ul,v is a hyperbolic plane and hence, by Proposition

13, it contains a vector ullsuch that (ul',ul'j = 0, (ul,ul')= 1. This proves the proposition for
m = 1.
Suppose now that m > 1 and the result holds for all smaller values of m. Let W be the
totally isotropic subspace with basis u2,.,.,u,. By Proposition 4, there exists a vector v E w - ~
such that (ul,v) # 0. The subspace H1 spanned by ul,v is a hyperbolic plane and hence it
contains a vector u l rsuch that (ulf,ul')= 0, (ul,ul')= 1. Since H1 is non-singular, H~' is also
non-singular and V = H1 I H ~ ' . Since W c H ~ ' , the result now follows by applying the
induction hypothesis to the subspace W of the quadratic space H~'.
PROPOSITION 15 Any quadratic space V can be represented as an orthogonal sum

where H I ,...,Hm are hyperbolic planes and the subspace Vo is anisotropic.
Proof Let V l be any subspace supplementary to v'. Then V 1is non-singular, by the definition
of V-L. If V l is anisotropic, we can take m = 0 and Vo= V1. Otherwise V 1contains an isotropic
vector and hence also a hyperbolic plane H1, by Proposition 14. By Proposition 3,

where V 2 = H~' n V 1 is non-singular. If V 2 is anisotropic, we can take Vo = V,. Otherwise
we repeat the process. After finitely many steps we must obtain a representation of the required
form, possibly with Vo= {O}.


1. Qundiatic spaces

Let V and V'be quadsatic spaces over the same field F. The quadxatic spaces V,V' are said
to be isometric if there exists a linear map cp: V + V' which is an isometry, i.e. it is bijective
and
(qv,cpv) = ( v , ~ for
) all v E V.
By (I), this implies

((~u,cpv)= (u,v) for all u,v E V.

The concept of isometry is only another way of looking at equivalence. For if cp: V -+V'
is an isometry, then V and V' have the same dimension. If u l ,...,u, is a basis for V and
ul',...,u,' a basis for V' then, since (u& = (cpuJ,cpuk),the isometry is completely determined
by the change of basis in V' from qul ,...,(pu,, to u L,...,
1 u,'.
A particularly simple type of isometry is defined in the following way. Let V be a
quadratic space and a vector such that ( w , ~#) O. The map z: V -+V defined by
MJ

is obviously linear. If W is the non-singular one-dimensional subspace spanned by w, then
V = W Iw'. Since zv = v if v E W' and zv = - v if v E W, it follows that z is bijective.
Writing a = - 2(v,w)/(w,w),we have

Tlws z is an isometry. Geometrically, z is a reflectioiz in the hypeiplane orthogonal to w. We
will refer to z = z,, as the reflection cor~espondingto the non-isotropic vector w.

PROPOSITION 16 If u,u' a m vectou of a quadratic space V such that (u,u) = (u',u') z 0,
then there exists a12 isometry cp: V + V sucl? that cpu = u'.

Proof Since
(U

+ u,',u + u') + (14 ul,u - u')
-

=

2(u,u) + 2(u',u')

=

4(u,u),

at least one of the vectors u + ul,u - u' is not isotropic. If u - u' is not isotropic, the reflection z
corresponding to MI = u - u' has the property zu = u', since ( u - u',u - u') = 2(u,u - u'). If

u + u' is not isotropic, the reflection z con-esponding to = u + u' has the propesty zu = - u'.
Since u' is not isotropic, the corresponding reflection cs maps u' onto - u', and hence the
isomehy csz maps u onto u'.
MJ

The proof of Proposition 16 has the following interesting consequence:


352

VII. The arithmetic of quadraticforms

PROPOSITION 17 Any isometry cp: V + V of a non-singular quadratic space V is a product

of reflections.
Proof Let u l ,...,u, be an orthogonal basis for V. By Proposition 16 and its proof, there exists
an isometry yf, which is either a reflection or a product of two reflections, such that v u l = cpul.
If U is the subspace with basis ul and W the subspace with basis uz,...,u,, then V = U I W and
W = u -is~non-singular. Since the isometry cpl = yrlcp fixes u l , we have also cplW = W. But
if o: W -+ W is a reflection, the extension z: V -+ V defined by zu = u if u E U , zw = o w if
w E W, is also a reflection. By using induction on the dimension n, it follows that cpl is a
product of reflections, and hence so also is cp.
By a more elaborate argument E. Cartan (1938) showed that any isometry of an ndimensional non-singular quadratic space is a product of at most n reflections.

PROPOSITION
18 Let V be a quadratic space with two orthogonal sum representations

v=

U I W = U ' I W'.

If there exists an isometry cp: U -+ U',then there exists an isometry v:V -+ V such that
v u = cpu for all u E U and yfW = W'. Thus if U is isometric to U: then W is isometric to W'.
Proof Let ul ,...,urn and urn+l,...,u, be bases for U and W respectively. If uj' = cpuj
(1 I j I m ) , then ul' ,...,u,' is a basis for U'. Let urn+l',...,u,' be a basis for W'. The
symmetric matrices associated with the bases ul,...,u, and ul',...,u,' of V have the form

which we will write as A 0 B, A 0 C. Thus the two matrices A 0 B , A 0 C are congruent.
is enough to show that this implies that B and C are congruent. For suppose C = StBS for some
invertible matrix S = (oii).If we define

ui'

=

",...,u," by

C J=rn+loji~j"
(m + 1 I i n),

then (uj1',uk")= (uj,uk) (m + 1 I j,k I n) and the linear map y:V -+ V defined by

is the required isometry.
By taking the bases for U,W,W' to be orthogonal bases we are reduced to the case in which

353

1. Quadratic spaces

A,B,C are diagonal matrices. We may choose the notation so that A = diag [al,...,a,], where
a j # O f o r j I r and aj = O f o r j > r . If al #O, i.e. if r > 0, andif we wliteA'= diag [a2,...,a,],
then it follows from Propositions 1 and 16 that the matsices A'@ B and A' O C are congruent.
Proceeding in this way, we are reduced to the case A = 0 .
Thus we now suppose A = 0. We may assume B # 0, C # 0, since otherwise the result is
obvious. We may choose the notation also so that B = 0, O B' and C = 0, O C', where B' is
s < lz - nz. If Tt(O,,+, O C7T = Om+,O B', where
non-singulas and 0 I

then T4CtT4= B'. Since B' is non-singular, so also is T4 and thus B' and C' are congruent. It
follows that B and C are also congruent.

COROLLARY19 If a non-singular subspace U of a quadratic space V is isonzetric to
another subspace U', then

u-' is isometric to u".

Proof This follows at once from Proposition 18, since U'is also non-singular and

The first statement of Proposition 18 is known as Witt's extension theorem and the
second statement as Witt's cancellation theorem. It was Corollary 19 which was actually
proved by Witt (1937).
There is also another version of the extension theorem, which says that if cp: U + U' is an
isometry between two subspaces U,U' of a non-singular quadratic space V , then there exists an

isometry y:V -+ V such that yfu = cpu for all u E U. For non-singular U this has just been
proved, and the singular case can be reduced to the non-singular by applying (several times, if
necessary) the following lemma.

LEMMA20 Let V be a non-singular quadratic space. If U,U' are singulur subspuces of V
- -

and if there exists an isometry cp: U -+ U ' , the11 there exist subspaces U , U ' properly
containing U,U' respectively and an isometry : 8 -+8'such that

Proof B y hypothesis there exists a nonzero vector ul E U n u'. Then U has a basis u l ,...,u,
with ul as fkst vector. By Psoposition 4, there exists a vector w E V such that
(ul,w)= 1, (uj,w)= O for 1 < j I m.


VII. The m-itlzmetic of quadratic forms

354

Moreover we may assume (W,MJ)
= 0, by replacing w by w - a u l , where a = (w,w)/2.If W is
the 1-dimensional subspace spanned by MJ, then U n W = ( 0 ) and V = U + W contains U
properly.
The same construction can be applied to U', with the basis q u l ,...,cpu,, to obtain an
isotropic vector w' and a subspace U' = U' + W'. The lineas map : V + 0'
defined by

is easily seen to have the required propesties.
As an application of Proposition 18, we will consider the uniqueness of the representation
obtained in Proposition 15.

PROPOSITION
21 Suppose the quadratic space V can be represerzted as apz orthogonal sun?

where U is totally isotropic, H is the orthogoml sun1 of nz hyperbolic planes, and the
subspace Vo is anisotropic.
Then U = v', nz = ind V - dim v', arid Vo is uniquely detern~inedup to an isometry.
Proof Since H and Vo are non-singulas, so also is W = H I Vo. Hence, by the remark after
the proof of Proposition 3, U = w'-. Since U c u', it follows that U c v'. In fact U = v'-,
since w n V' = {O).
The subspace H has two nz-dimensional totally isotropic subspaces U1,Ulfsuch that

+ U 1 is a totally isotropic subspace of V. In fact VI is maximal, since any
isotropic vector in U1' IVo is already contained in U1'. Thus nz = ind V - dim v-' is uniquely
Evidently VI: = V'

determined and H is uniquely determined up to an isomet~y.If also

where H' is the orthogonal sum of m hyperbolic planes and Vo' is anisotropic then, by
Psoposition 18, Vo is isometric to Vat.
Proposition 21 reduces the problem of equivalence for quadratic forms over an arbitrary
field to the case of anisotropic forms. As we will see, this can still be a difficult problem, even
for the field of rational numbers.


I . Quadratic spaces

355

Two quadratic spaces V, V' over the same field F may be said to be Witt-equivalent, in
symbols V = V', if their anisotropic components Vo, VO1are isometric. This is certainly an

equivalence relation. The cancellation law makes it possible to define various algebraic
operations on the set W(F)of all quadratic spaces over the field F , with equality replaced by
Witt-equivalence. If we define - V to be the quadratic space with the same underlying vector
space as V but with (vl,v2)replaced by - (vl,v2),then

VL(-V)= { O } .
If we define the sum of two quadratic spaces V and W to be V I W, then

Similarly, if we define the product of V and W to be the tensor product V €9 W of the underlying
vector spaces with the quadratic space structure defined by

then

It is readily seen that in this way W ( F ) acquires the structure of a commutative ring, the Witt

ring of the field F.

2 The Hilbert symbol
Again let F be any field of characteristic + 2 and FX the multiplicative group of all nonzero
where a,b E F X ,by
elements of F. We define the Hilbert symbol

( ~ , b =) ~1 if there exist x,y

E

F such that ax2 + by2 = 1 ,

= - 1 otherwise.

c2

By Proposition 6, ( ~ , b=) 1~i f and only i f the ternary quadratic form a t 2 + b q 2 - is
isotropic.
The following lemma shows that the Hilbert symbol can also be defined in an asymmetric
way:


356

VII. The arithmetic of quadratic forms

LEMMA 22 F o r anyfield F and any a,b E F X ,( ~ , b=) 1~ if and only if the binuty quadratic
form f, = 5 2 - aq2 represents b. Morover, for any a E FX,the set G, of all b E FXwhich
are represented by f, is a subgroup of FX.
Proof

Suppose first that ax2 + by2 = 1 for some x,y E F . If a is a square, the quadratic form

f, is isotropic and hence universal. If a is not a square, then y # 0 and (j-l)2 - a(xylj2 = b.
Suppose next that u2 - av2 = b for some u,v E F. If - ba-1 is a square, the quadratic form
at2 + b q 2 is isotropic and hence universal. If - ba-l is not a square, then u # 0 and
a(vu-1)2 + b(u-l)2 = 1.
It is obvious that if b E G,, then also b-1 E G,, and it is easily ve~lfiedthat if

then

(In fact this is just Brahmagupta's identity, already encountered in $4 of Chapter 1V.j It follows
that G, is a subgroup of FX.

PROPOSITION
23 For any field F , the Hilbert symbol has the following properties:
(0 (a,b), = @,a),,
(iij ( a , b ~ 2=) ~(a,b), for any c E F X ,
(iii) (a,
= 1,

(iv) (4- ablF = (a,b)F,
(v) if(a,bIF = 1, then ( ~ , b c =) ~
(a,cjFfor any c E Ex.
Proof The first three properties follow immediately from the definition. The fourth property
follows from Lemma 22. For, since G, is a group and f, represents - a,,f, represents - a b if
and only if it represents b. The proof of (v) is similar: if fa represents b, then it represents bc if
and only if it represents c. 0
The Hilbert symbol will now be evaluated for the real field [W = Q, and the y-adic fields
Q, studied in Chapter VI. In these cases it will be denoted simply by (a,b),, resp. (a,b),. For
the real field, we obtain at once from the definition of the Hilbert symbol

PROPOSITION
24 Let a,b E RX. Then (a,b),

= - 1 if and only if both u

< 0 and b < 0. El

To evaluate (a,bjp, we first note that we can write a = p W , b = ppb', where a,p E Z and
la'/, = lb'lp = 1. It follows from (i),(iij of Proposition 23 that we may assume a$ E { O , l }.


357

2. The Hilbert symbol

Furthermore, by (ii),(iv) of Proposition 23 we may assume that a and P are not both 1. Thus
we are reduced to the case where a is ap-adic unit and either b is ap-adic unit or b =pb: where
b' is a p-adic unit. To evaluate (a,b)punder these assumptions we will use the conditions for a
p-adic unit to be a square which were derived in Chapter VI. It is convenient to treat the case
p = 2 separately.

PROPOSITION
25 Let p be an odd prime and a,b E Qp with lab = lbip = 1. Then
(9 (a,b), = 1,
(ii) (a,pb), = 1 if and only i f a = c2for some c E

Qgp.

In particular, for any integers a,b not divisible by p, (a,b)p= 1 and ( a , ~ b=) (alp),
~
where (alp) is the Legendre symbol.
Proof Let S c Z p be a set of representatives, with 0 E S , of the finite residue field
Fp = Zp/pZp. There exist non-zero ao,boE S such that

But Lemma 11 implies that there exist xo,yo E S such that

Since Ixolp 2 1, lyolp 4 1, it follows that

Hence, by Proposition V1.16, axo2 + byo2 = z2 for some z E Qp. Since z f 0, this implies
(a,b& = 1. This proves (i).
If a = c2 for some c E Q,, then (a,pb), = 1, by Proposition 23. Conversely, suppose
there exist x,y E Qp such that ax2 + pby2 = 1. Then lax2Ip# bby2Ip,since lalp = lblp = 1. It

follows that IxL = 1, lylp I 1. Thus lax2 - 11, < 1 and hence ax2 = z2 for some z E Qpx. This
proves (ii).
The special case where a and b are integers now follows from Corollary VI.17. 0

COROLLARY26 If p is an odd prime and if a,b,c E Q p are p-adic units, then the
quadratic form at2 + bq2 + cC2 is isotropic.
Proof The quadratic form - c-lac2 - c-lbq2 Proposition 25. 0

c2 is isotropic, since (- c-la,

-~

l b= )1, ~by