Vector spaces of countable dimension over algebraic number fields
by Leon Eugene Mattics
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Mathematics
Montana State University
© Copyright by Leon Eugene Mattics (1967)
Abstract:
An investigation of denumerably infinite dimensional vector spaces over algebraic number fields
supplied with non-degenerate Hermitian forms is undertaken. The Hermitian forms are divided into
Glasses according to the underlying field automorphisms. If Κ is a field and ς is an automorphism such
that(Formula not captured by OCR)then &sigman; is called weakly orthonormal (orthonormal) if and
only if every denumerably infinite-dimensional Κ -vector space supplied by a non-degenerate form out
of the class of Hermitian forms of ς has a(Formula not captured by OCR) orthogonal basis (an
orthonormal basis). A field &Kappa is called weakly orthonormal (orthonormal) if and only if
(Formula not captured by OCR) is weakly orthonormal (orthonormal). Finite extensions of orthonormal
fields are orthonormal fields.
By the Hasse-Minkowski theory of quadratic forms, the algebraic extensions of the rationale which are
orthonormal are precisely the non-formally real algebraic extensions. The weakly orthonormal
algebraic extensions of the nationals are precisely the algebraic extensions with at most one ordering.
Over algebraic extensions of the nationals with precisely one ordering, Sylvester's theorem holds for
denumerably infinite-dimensional vector spaces supplied with symmetric non-degenerate bilinear
forms; so, such spaces are characterized by a complete set of two cardinal invariants.
Let ς be an arbitrary automorphism over algebraic number field Κ with(Formula not captured by OCR)
Criteria, depending on the fixed field of ς in Κ and the generator of KL over the fixed field of ς, are
given to determine whether or not ς is orthonormal (weakly orthonormal). J
VECTOR SPACES OF COUNTABLE DIMENSION
OVER ALGEBRAIC NUMBER FIELDS
by
LEON EUGENE MATTIC S
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements for the degree
of
'.DOCTOR OF PHILOSOPHY
‘
in
Mathematic s
Approved:
ead, Major Department _
I-OSS Chairman, Examinini
Committee
an. Graduate Division
,1 MONTANA STATE UNIVERSITY
; Bozeman, Montana
June, 1967
(ill)
ACKNOWLEDGMENT
The author is deeply indebted to his teacher and thesis
advisor, Dre Herbert Gross.
(iv)
TABLE OF CONTENTS
CHAPTER
I.
II.
HI.
IV.
PAGE
INTRODUCTION
I
ORTHONORMAL FIELDS
5
ORTHONORMAL ALGEBRAIC NUMBER FIELDS
10'
HERMITIAN FORMS
22
LITERATURE CITED
30
(v)
ABSTRACT ■
An investigation of denumerably infinite dimensional vector spaces
over algebraic number fields supplied with non-degenerate Hermitian
forms is undertaken. The Hermitian forms are divided into Glasses ac­
cording to the underlying field automorphisms. If K- is a field and
CT is an automorphism such that
<s" o cr ■=
then cs~ is called
weakly orthonormal.(orthonormal) if and only if every denumerably in­
finite-dimensional
K- -vector space supplied by a non-degenerate form
out of the class of Hermitian forms of cr~ has
a j=.
ortho­
gonal basis (an orthonormal basis). A field K_ is called weakly
orthonormal (orthonormal) if and only if " H vl is weakly orthonormal
(orthonormal). Finite extensions of orthonormal fields are orthonormal
fields.
By the Hasse-Minkowski theory of quadratic forms, the algebraic
extensions of the rationals which are orthonormal are precisely the nonformally real algebraic extensions. The weakly orthonormal algebraic
extensions of the rationals are precisely the algebraic extensions with
at most one ordering. Over algebraic extensions of the rationals with
precisely one ordering, Sylvester's theorem holds for denumerably
infinite-dimensional vector spaces supplied with symmetric non-degenerate
bilinear forms;
so, such spaces are characterized by a complete set of
two cardinal invariants.
Let cr*. be an arbitrary automorphism over algebraic number field K.
with <5— a <3- -= "!SLx*. . Criteria, depending on the fixed field of
in K- and the generator of K- over the fixed field of
, are given
to determine whether or not cr is orthonormal (weakly orthonormal).
CHAPTER,I
INTRODUCTION
Let VC be a commutative field and cT
an automorphism of K- such
that the composite map c r <=»<$" is the identity automorphism,
\C we shall write ^
VL
.
If <=*< is any element of
a
_
({<1 ,<5^) -vector space (SE,
over VL
X
Xyr--Yf '
.
together with a bilinear form H " G X E
VA
^ IA
1-1 (x, ^ )
- H
on
we shall mean a vector space
— -z (-C-
xZ^) , V
^ 6 K
^
, V
y ;N) 6 E
^
(^i v )
. . By
such that
El,
We shall call such a form a Hermitian form over VL with respect to <5~
and denote the class of all such IA by ( 1<L ^<7" *)
subspace of VZ
with the form
? %>y ( V-%
A
F ^ x P" »
A
K- -vector space E
is non-degenerate.
write W y W - V - I C X j Y s)
Two vectors X
H L X 1nZN
O
and Sj
^
together
It is easy to show that any
defines at least one Hermitian form
.
(K. ic^) -vector space (
the form |A
is a linear
If there is no risk of confu­
instead of (,£■ A').
automorphism c$~ with <5~ o d ' =. ILjc
over a
If F*
we shall mean the subspaoe
restricted to
sion, we shall write £
*
H ) is called semisimple if and only if
Given (,Ey
and call
in a space
and Y <5 Lr
then we shall
\\>C\\ the length of the vector X
•
(Ey Vi) are orthogonal if and only if
• We shall take the usual meanings for orthogonal basis
and orthonormal basis.
If the (algebraic) dimension of ( E 7 VL) is denumerably infinite, we
shall write
cltw\£ - ^ o
or
CGTjLAs)"=
of C E A) is at most countable, then we know that £
•
If the dimension
has an orthogonal
2
basis with respect to H
•
Attempts to classify spaces (
up to
orthogonal automorphism have-, for this reason, been restricted to the
cases of finite and denumerable dimensions.
Again let cf
<r o cT =.
every
.
be an automorphism of a commutative field \<L with
We shall say that <f
-dimensional semisimple
orthonormal basis.
field VC
If
is orthonormal if and only if •
\C. -vector space with H 6
has an
is orthonormal then we shall say that the
is orthonormal (i.ei,
1-4. is orthonormal if and only if. every
- dimensional vector space (.E. j §>) over VC with a symmetric bilinear
form has an orthonormal basis).
normal is and only if every
( E 1W ) with
such that
We shall say that df* is weakly ortho­
-dimensional semisimple vector space
admits of :an orthogonal decomposition, E - F" A
has an orthonormal basis and Gr
basis (i.e., an orthogonal basis ]
with
Gr-
has a negative orthonormal
=
for all I).
If dL^is
weakly orthonormal, then we shall say that \C is weakly orthonormal.
In this paper we shall investigate the problem of determining the
orthonormal and weakly orthonormal automorphisms of algebraic number
fields.
,
This investigation is initiated in Chapters .11 and III with the
study of orthonormal and weakly orthonormal fields.
In Chapter III we
shall find a surprisingly easy answer (Theorem 3 and Corollary 8) to the
classification problem for orthonormal and weakly orthonormal algebraic
number fields (a problem proposed by Kaplansky) and we shall even be able
to extend the classification, by the Hass e-Minkowski theory of quadratic
forms, to all algebraic extensions of the nationals.
- 3 In Chapter II we shall show that finite algebraic extensions of
orthonormal fields are also, orthonormal fields (Theorem l) by use of
the following theorem of Fischer and Gross:
Theorem Gi
(
» Corollary 4, pp.23Q)
real field with char K. "A
dimensional
Let ^
2-.
be a not formally
§i) is an
.If '
-
K r -vector space with a non-degenerate, sym­
metric bilinear form
<j) then ( . E T ^ admits of an ortho-
normal basis if and only if there is an
linear subspaoe p-
of E,
such that
-dimensional
^jprC F"VC
^
^ O % .
The classification problem (as regards orthonormal and weakly ortho­
normal fields) has a simple answer for large classes of underlying fields.
This is best illustrated by a theorem of Kaplansky:
Theorem Kr
Let
^e a field (char
some fixed
space
, every
2- ) with the property that for
-dimensional semisimple vector
(EI4^ with <4€ C ;dL
has a vector of length
Ar i or — 1
IL then KL is a weakly orthonormal field.
if
K»
is not formally real, then
K-
Further,
is an orthonormal field.
The proof of the theorem proceeds by inductive construction.
A
critical role is played by the fact that, under the conditions stated,
one can find "enough" subspaces of finite dimension which are spanned by
orthogonal bases with all basis vectors of the same length.
In Chapter III we shall prove a theorem (Theorem 2) which guarantees
the existence of such subspaces in the case of an arbitrary (denumerable)
space (Ejtjp) over an algebraic number field*
(such as K
For special choices of
K-
not formally real) one obtains an elementary proof for the
existence of orthonormal bases;
however, the Hasse-Minkowski theory of
quadratic forms (see C"il -^or a^1 exposition) yields much stronger results
so the usefulness of Theorem 2 will not be demonstrated until the study of
arbitrary Hermitian forms is undertaken in Chapter IV.
In Chapter IV necessary and sufficient conditions are given to
determine whether or not an arbitrary automorphism of an algebraic number
field is orthonormal or weakly orthonormal.
Several examples are then
given to demonstrate the use of this criteria.
Notation:
In Chapters II and III, we shall study symmetric bilinear forms
exclusively.
the form
Thus, if we write
is symmetric.
we shall automatically mean that
If we write ( E jV^x) we shall mean that H
is
an arbitrary Hermitian form.
If ^i\—
E <£; _fS—
is an index set and
then by
the F h tS «
By
E -
E =
is a subspace of E
we mean that E
6
decomposition of the spaces
-
we mean that
^ -fc.£ _ A — .
E
for each
a direct sum of
isan-orthogonal
This and all other vector
space notation is consistent w i t h L / O *
We shall always denote
thefield of rational numbers
K ( © ) is a simple algebraicextension
K(6) then X
of K.
by
. If
with [ K ( 8) I\<3 - Vi
has a unique representation as a polynomial in Q
degree less than Vl
.
We denote the degree of this polynomial by
All field theoretic notation is consistent with LtO-] .
and
of
.
CHAPTER II
ORTHONORMAL FIELDS
Some examples of orthonormal fields are the complex numbers
the algebraic closure of the rationale G)
in C.
C ,
and any finite field.
All of these fields have the property that the order of the multiplica­
tive group of the field modulo square factors is finite..
Such fields are
called Kneser fields and were found to be orthonormal.fields in C.2-1 .
It is quite obvious that all finite algebraic extensions of the
fields in the examples are also orthonormal fields.
In this section we
shall show that any finite algebraic extension of an orthonormal field
is orthonormal.
Corollary I :
(To definition)
then
Proof:
If a field K*
is non-formally real.
By the definition of orthonormal there is at least one positive
Oa
integer
K
.
is an orthonormal field,
such that the equation
Thus,
%
X.
—
has a solution in
VC is non-formally real.
We shall now list several corollaries of Theorem G.
To avoid
monotonous repetition of the same sort of argument, we shall only give the
proof of Corollary 6.
Corollary
2:
A field
IfC
is orthonormal if and only if every semisimple
(Vg-dimensional
|C -vector space possesses an infinite
dimensional totally isotropic subspace.
Corollary 3 :
A field VC is orthonormal if and only if every semisimple
/^-dimensional
vector
\C. -vector space has a non zero isotropic
-SCorollary 4 }
Let ^
be an orthonormal field and t.Q
of non zero elements of
'
Let
Then there are elements
with
such that
a SL/ * '
^ CK^
Let
Then there are elements
in the sequence and elements
in the field l<. with
elements «=><•* . • - <=*-
*-)
that
L'?
L Z 1_
re- rn
'
J
x
•
prove the necessity,
the field with
o
be an
in the field ^
with
<D
such
.
" L "L
-
“ ■ **
It remains only to
there exists -=^1.) • • - - = ^ vvi in
and such that
— -j_ , i.e.
(/^-dimensional semi simple
has an orthogonal basis, say
we have that
non zero elements
We notice immediately that K . is non-formally real,
since for the sequence i
O
each L .
^
.
— - I. .
\<- -vector space then
^
Since (..Ej <^) is semisimple,
Therefore, consider the sequence
By hypothesis there are elements H <25.11 * ' ' .U^. j\in the
sequence and elements
in the field such that
We now consider the sequence
^
X. WNJr'!.
)' ' '
.
in the sequence and
The sufficiency follows from corollary 4.
i'i.'L
-= O
be a field then \C is an orthonormal field if and
of VC. there are elements Qii
Let (j- t
* '/0 ^
such that 7 ^<?i
q
only if for every sequence
Proof:
.
Ck ■
be an orthonormal field and £<3, ^ ^ ^ be a sequence
of non zero elements of K. .
Corollary 6 :
^ ^ be a sequence
in the sequence and elements o f ^ • *
in the field
Corollary 5 :
,
.
t
l\S
and elements, of the field
.
By hypothesis we find
* * ' T fWNV^
-7such that
^e''-^C><0 —
"I* or
^ evIL ~ "
We now have that
•s” WVvfJj
the non zero vector
is isotropici
By Corollary 3 the
assertion is proved.
We shall use the corollaries to obtain the main result of
Chapter II,*.
Theorem I ;
Let VC* be an orthonormal field and
extension then V v C©)
Proof:
Assume CK(.©V. t O - W V .
O -dimensional
CE, ^
•
is orthonormal.
Let
I1V ^ -vector space.
non zero isotropic vector.
ICCs'i a simple algebraic
Let
be an arbitrary
We must show that LEj
contains a
L*^ I. 156 an orthogonal basis of
By relabeling the subscripts of the <3 ^ S
we decompose (,Ej ^
1
in the following manner:
1)
Split - ;(lEL , <|,)
civmE^
2)
- C^o f°r '\jz. I. ♦
Note that
E
(,
,
^
E 1-^j
)
with the
is semisimple.
Split each ( E c ; <|>) into packets: ( £ I 1^ ) -
(Lvvl E c^z:
m)
into packets: C
for ^ *> IL »
Split each
Note that
• • ss. ^
(b.
with the
is semisimple.
( Irvt-I. subscripts) into packets:
Iafif) withthe
for p
F ix an
Step I)
-I. .
Note that ( E i. ^ . . >■*
Sl • > ‘ 1S SL
| <4 )
.
We assert that each vector space (
a non zero vector X
semisimple.
such that
^ Q tW X V W
.
E
.
— 2-
.
contains
8
-
-
Suppose that this is not the case.
that
= W i - IL
Then there is a p' "> zL
for any non zero vector 1X
such
in (E.
This space is semisimple and it has an orthogonal basis;say
Now
s
“
"^q C
U
with O, • •
«
. jf-o
= Vvt - -3-
.
Therefore,
K- .
T h u s,
\<—
.
O. jv \-i.( u 4 O
L
... • <s>^
in K» with
•= °
We have
i-/^ 4 o
•
i Z- <L. .
such that 3 © O U s .
in the
such that
since
^4"- O
has a non zero
.'SAy, 4 ^
~
and
contrary to the supposition.
.
^
' .
"4"v"^ ^
V ‘U
v. zL
‘
O
Now
^ A
Therefore, for. each
vector
fo r each
’
A
^ L1W w ^ cV.
and by supposition
is a sequence of non zero elements in
By Corollary 5 there exists <X^
sequence and of .
.
W X - Q L and thus
(E
:A C
contains an infinite set of linearly independent mutually orthogonal
vectors
Step 2)
^
^
^
Now let L^L""' >
that for each ^
remain fixed but vary JL .
there exists an infinite set of linearly independent
mutually orthogonal vectors
such that
We have from Step I
contained in
( \\£ js.^ w'X *£ W
\
"2-
•
••s
cji ^
We consider the following cases for
each 9. . .
Case i)
For one
p
,
ftp W
— O
then we are done, since
then contains a non zero isotropic vector.
Case ii)
One of the <Sa p, is such that d B U l Ce.
vector and call it
Case ill)
For every
.
< Wr2.
Choose this
*
we have that <^6 (U€ft^\\) ^ Wi-Z •
method of Step I) on the leading coefficients of
Reapply the
l\^ ,p, ^ to find a
9
non zero vector X
in the space generated by
is a subspace of
^ e. t 4> )
We call this vector
^
) such that
^
(which
^ to( U K U s)^ »^-2. .
«
Therefore, from Cases i), ii) and iii) either (Ej
isotropic vector or
.. -Sj
has a non zero
has an infinite set of mutually ortho­
gonal linearly independent vectors
CL
with
„
Continuing the process (if we don't find a non zero isotropic
vector at one step)
we find that (.Ej
has an infinite set of mutually
orthogonal linearly independent vectors
< 5 UU; -
Q
(i.e.
we find that ^ ^
But
IlU i1W
KL
).
*2 VA C.lc.-s' I. such that
Reapplying the method of Step I
has a non zero isotropic vector.
was arbitrary, therefore the theorem is proved by
Corollary 3.
Theorem I immediately leads to;
Corollary 7 :
If Vts is an orthonormal field and
extension of
Remark;
^
Vs. such
-dimensional semisimple
tropic vectors and
is a finite algebraic
, then \<L, is orthonormal.
Gross has asked the following question;
orthonormal field
VL
Does there exist an
that for any integer Vl > £> there is a
K. -vector space
|\j< chin
F<
(^\ with no non zero iso­
?
If the answer is in the negative, then Theorem I has a much simpler
proof.
CHAPTER III
. ORTHONORMAL ALGEBRAIC. NUMBER FIELDS,
In chapter II
we studied fields in general;
in this chapter we
turn our attention to algebraic number fields.
is a finite algebraic extension of Q
If K
a perfect field, there is an element O
Furthermore, if <& — <Q.~ ^
Definition I ;
Let N
and
§ (M,vvt - 2.*
with
of
Q
then because
K. such that
then
(o") —
K
- Q
is
(Q) „
C5>(_0)
.
ITl be two positive integers then we define
^ 3 VzV4r-V<lV H - 3
' H > I- j
- S ( U - - I ) -V I.
Theorem 2 ;
Let
be a finite algebraic extension of the rationale
with
CG^C©) .
— VX .
Let
VA be an arbitrary posi­
tive integer then any semisimple Q / S'”)-vector space
of dimension at least ^ ( W ,
contains M
linearly inde­
pendent mutually orthogonal vectors of the same length.
Furthermore, the proof of the theorem will show that we may select
the VA
mutually orthogonal vectors such that their lengths are not zero.
Before proving this theorem, we list several lemmas.
If
|”
I©V
is a finite algebraic extension of
— YX , then any element
i
tation as a polynomial in &
with coefficients in Q
.
of
C
with
has a unique represen­
of degree less than or equal to VX — {L
Weldenote by
the degree of this
polynomial.
Lemma I:
Let Q
C
be a finite algebraic extension, of Q
L Q C © V . Q l - VX .
If
and
Q - B t* c- Q Ce) with Q vvx Al O
11 and VY\■< V\ then there is O
i) Q i e ' t - Q C m
3) C t- * O
,
If w\
and
O vvt
= Z j
Thus, the coefficients
"2-
C
J for ^7 G
We take
the first terms of each polynomial in
C^ .
Lemma 2 ;
.
p > 0
;
•
are polynomials in
efficients of the same sign as A ^ ,
- <s> — p
i -£
ifI'X
w>-:
a
<D
O we take 0 - ^ . Q
If VVX > O then we set
p* P
pyV
_m
• ; Iw%
^•E CVu G c •=: 5 - a. 'u ( © ~ p-V- p 1) s
S
such that
Vn _ —
2)
Proof of Lemma I:
in
p
p
p
with leading co­
large enough so that
are dominate.
Then for such
Further,
» i.e. we set
.
Let Q (.Q n) be an algebraic extension of <Q
Let Q ^ » » t^SL,
be elements of
f o r each i ( i t .
b^x
Furthermore, assume that O i
uyrk^ O
Sv^YV b
SX<^yx
with VjQlGx)! <j^ —Y\ •
Lo") such that
C-tP-') and
•
( I t L t A-j and that
. for I t L , ^ t
;
then there
exists 0 <£ (^ (©*) such that
I)
Proof of Lemma 2:
■=. Q L o ) ,
Z T t f c 1C U W - G ^ j- C Lfct O
V Z - T J
l
3) s
W ^ vvk ^
We set ©
5 1^VV
— ©-pas
for all L ^
in the proof of Lemma I.
follows from that proof that we merely have to choose
It is, of course, essential that JL< O O ,
p
all S t e j
.
It
large enough.
- 12 Definition 2 »
Let
be a finite algebraic extension of
CQteY. Q 3
.
Let X e Q t e xI ,
be an integer such that
that X
C> I
<IA.
has an induced degree
with
» and let VVl
Then we shall say ,
(denoted
~
if and only if there is an element =Y <= Q
)
such that
rT v v ^ .
Remark:
Lemma 3 :
We note that X
Let
Q
may admit various induced degrees..
( O') be a finite extension of the rationale with
~
either
Vl y
I..
If
X^
y
Q>(.©\ with
- Y l - I L or ^
^
o
We may have
- V l - 2L.
Proof of Lemma 3:
We can always achieve this by multiplying X
suitable power of
O
by a
.
We now state a theorem proved by Minkowski (See C"IfJ * p. 433).
Theorem M ;
Given rationale 0 ^ - 0 , Io 4 O »
that the form
- V - V
O
C
« &■ 4-0 * <2.4 O 4 such
4
is indefi­
nite, then the form has a non-trivial zero in the rationale.
We now use Theorem M and the proceeding lemmas to complete the proof
of Theorem 2»
Proof of the Theorem;
If
Irl — 3- then by Theorem M
^(VA1yL'')
is indeed
sufficient, so let us assume that Yi > •!. .
(E;
IcX
,
ujIifsgov*
is semisimple, so it has an orthogonal basis
with
4 O
.
Therefore, by multiplying our basis vectors
by suitable scalars
I ^
— 3
G. Q
we have by Lemma 3 that
for each L
.
Call this new basis
:
^
l.
In")
( <2 ^
Xe
)•
“ IJ We note that:
From this we deduce that at least one of the following cases must
hold:
Case I)
There are at least Z - ' ^
^€[^3
^
Case 2)
^
^
- V X - I. vectors in the basis
whose lengths have a degree in ©
There are at least 2.,>!v
•. *W
over
Q i of v\ - I. .
jL vectors in the basis
< L -6.^UjlIv^ wllose lengths have a degree in
£=► over
<Q
of \r\— 2. .
From this we may deduce that at least one of the four following
cases must hold:
Case la)
There are at least zT
vectors
G.
in the basis
i
"[g ^3
< I* ^UXlI-Vvllose lenBtI1S (as minimal polynomials in ©
degree
r\«=l. and positive leading coefficients.
) have
ArVX-3
Case lb)
There are at least 4"
» V\
vectors in the basis
^ ^ ^ . ^ ( V X 1VO) whose lengths have degree
— 3- and negative leading co
efficients.
A Xa Case 2a)
There are. at least
•M
vectors in the basis
whose lengths have degree
^
and positive leading co
efficients.
Mvx--IVjS
Case 2b)
There are at least 4-
^ G ^3^A
•K
vectors in the basis
w^lose lengths have degree r\- %. and negative leading
coefficients.
Assume that Case la) holds;
b = = is I E lI l 5
=ay
then we may choose vectors out of the
i. 4,
such ‘"at a L|n_1><3 .
1:-5l L-C ^ rn' 3- N
.
Call
the subspace generated by these vectors
4_«. L* 4^"
^
.
Note that
atl orthogonal basis and that (Fj
By Lemma 2 we may change to a new generator 0
with
and each
^
L
for each
with L ^
with
is semisimple
of .G?l®V
such that
! . < ( . * ^ vi"3.
.
We divide the basis ^ ^
t i V y t i L , S4 ^ 2-,
2, ,
^
^ C,M "cVa w ^ vA into 4"
v
43 ■
exist rationale, % ^ +<u . * * 4 ^ <2. *
disjoint packets
For each packet, by Theorem M, there
* not a11 zero 8Uch
^
that
- I-
4
-
I L ^
vi:
Then we define. -V. “ zi ^ ^ ck", . v"^r. ?.* for each
I-=
X1
1 4-^ ^ ^ ^ ^
independent vectors.
I
^ I. < ^
is a set of mutually orthogonal linearly
Define
^
basis
•
e
to 1)6 t^e space generated by
Then
Also for each
^
( n _1_-1 and V \ ^ > ’ O
with
with
-H .
4
Then
^
^
is semisimple with orthogonal
we have that
for yL ^ (k. < n - IL .
Now we try to continue the process.
Suppose we have already found
vector spaces
p
c p
> «A— V<-
and
VA — Vt--VIL
has the following properties:
L) dxvwv F vx_^ = Ar
CFvi.
Vx-VU1
.M
has an orthogonal basis
•vx-v
a) \\U •\\—
v
and lFn-%. I
)4^^ is semisimple.
4 (.V1
T 1I ">
L lA
4'
-A
'0
i
i- Z" ^
Li © ^
iJ
-uuv
VNs-v
Vu
U
6
^or
(-I I-
such that:
.A (y\-A
L^A
* (A ,
,
■
- 15 -
b) X; > O for all ^ ; r\—
,
c) C
■ > O
I
for all
Assumlng that
^ V'x— IL t
V^-V-sI
t
M
, (L -
and
v^-Vd-
« . < y\ , we shall construct
the same properties as F
1 h-w.
. W e
F vxw^ w i t h
study first the
Case IaA). Suppose there exists a set
t u j
1.1 (2.-LH
such that C
V \- XC - L
^ . i. < V r l v i H
for all
L wxYi-VI -
W ^^
with 1.1 VvV1 fi. «- \/\
A (Vt-\6and
v 4-
^
' * Kl
.
Then select any A v^
» M.
of the U- 'S »
tA
say
1_1 IL
^
and let them generate the
r~
r
a
.
-
vc-
I. has the required properties;
A A_
C .n _y.. i.
.
4.(.a -v -1\
,
'kl),
If not Case IaA)„ then Case IaB).
Since we shall worry only about the coefficients C.
C Vl- V^- fixed) we shall put C , :
^sC.
U I A - VL- I.
u
.
,^
Vi--O-
Clearly either (I) more than
or exactly one half of the C ^ 1S are greater than or equal to A ^
(II) more than or exactly one half of the
to - K - K
; (1 1 L
C >
S
or
are less than or equal
4 4'"'",'
S. M I ;
Without loss of generality, assume that (I) holds.
I
^
(and hence the corresponding
Xv>-K. ^ C 1 ^
II 1
u-K S
We relabel the
), so that
* 4: C- (.^UA-VC-Xu ') /%_
Since we are not in case la), and since
2. . ^ . V lv'"‘‘" :L\
N
I
w
u ^ IO-
-
M
ArVvx- Id-there are at least %.0,Ac
greater than C.^ t
- Ui-i)^.
the one above, relabeling CN *
Ni
of the
which are strictly
Thus, we may construct a subsequence of
is necessary (and hence the corresponding
- 16 LA ^ 1 S
)j such that:
• • 1 - c i» ■
-
By the Archimedian Property of the rationale, we choose a positive
rational
p
such that:
W
Now let O
- B
G / p then
the .2-0
O i e ) (=QCS1)= Q i S ' ) ' ) ..
1"-*-1-'.N Vi5 " ' " c to
0.^0
-
in
&
We have
2 : ^W-\<_
" ^ A ;^ X
U V W - Z
c . .Q 1
j<.v\*vc
We now form five-tuples of elements of. %
^ A 20»
^
in the following special way.
A (Vt-V-•
For each
with
1.6. L ^ ^;*4
.A-Cna -V--IlA
1S
with 4 " 4r
U l7,* U i^
4'4-
X<
•
.
,
-V-*L
«W
we choose four distinct
4 vX
^ t. Q.O ‘ 4"
‘H
;
in this way. we divide. Ju.} u
mutually disjoint packets
call them
£ o .^ U -
k -i S
W U iiW = x
4-4«.
. ,
2:
-v*c .
.
p
c.-p^v
”
^V-V.'!.
u1
1 jA
H - V - - 1L.
vith C . ^
P w ' ^ Vx-X^
ay our construction.
f
P vx"11 Avx-^^-
and c Lk f “" V
So we now have that:
> O
P vx- ^ --11-«=f O
,
;
^
into
^ L1 ^ u L I. v t^ c-Z. j U CAr Ii
Now
(iu.i\=i\.
,
( ! < - ( : < Ar),
^ ^
I
-17Therefore4 by Theorem M the quadratic form
t-o
has a non-trivial zero in the rationale;
say
f tV-/ g. ^ ^
-6-4" ^
Let O v. =; "% c^- ,
'
then we have that
^
a
^ < P
- V ^ =- < C ^
+ J T ^ U c ^l
We define XI1= cV. U .
^ [J. ?
*■ L' 4-i.
4r«4-
/,
independent vectors out of
^
6 e-vL
for each Z
.
Then
^ and
Pkx"
^V
-
is a mutually orthogonal set of linearly
4 t-A
\\VA^\\ crrcvZ
.
+
V
%-
p N . . 5>^
;-CVN-V-1 V
U
6 = 4with "I 4=. C *h
•4
'N and d • - - 5^
N
,
., 4 ZI <=<~., C . , .
LD
Changing back to the generator &
-6= 1 ^6
of QtE^) we have that
VN-I
IlU-W -0T-^21 ^n-G
L
*■ vv-v.
»
Define
‘
'
^ to be p
have cl,• -> o
*- \
llu.ll=
-©yN - * . ^
4r ?^iri-\<.
for all £. I .
*a
^
;
^
2Z
^ Ll G jf
i
^w-Vc-I.
by our induction assumption c)
Therefore
ZL A e j +- JEL. <j..Gij" Xl^ i.+ 4- +v^
"-"'I
»
X
> 4-VM-V--1.') ^ 4
^rCvN-16,-<\
• ^n\
.M^ into 4"
packets of
i , ^,xtNA- +
We have;
tin.IAuX-VtrvW- 4V'tAc)+t
'SlXiSj + 2 1
,
VN-Vu--J A "
’ ^ vn-V--IL
Nowcv,.v
solution
>o
■N .
1
We finally s p l i t l u j
d .. .. 6 1 , ( i "'MiXvt7^'1
' /
for 1-t t 6.4- "so by Theorem M there is a
\ ^ i.&)
we
in t^e rationale to the equation
- 18 —
I -
X'
e-! -*<
then
*=2 V
t*\— I
« u \ . i v « r X 1S ^ + z
m
with V>X . • > O
>. -».-i i
"-a
_
Denote the space generated by theftA.S
t v,
....
'T h M R - V 1- V c
properties
a“d ' • ■b-wri,
L) and
tL j
with basis
.
JCt^ ^
by F n . ^ L
satisfies
with j O 1_ in lieu of k- .
By this construction we have proved the theorem for Case la).
The
proofs for Cases lb), 2a) and 2b) are nearly the same, word for word, and
are omitted here.
Remark:
This concludes the proof of the theorem.
A theorem of the same nature as Theorem 2 can he derived from
the results of the Hasse-Minkowski theory.
With Theorem 2 we could .
classify all orthonormal and weakly orthonormal algebraic number fields.
However, we shall use the results of the Hasse-Minkowski theory to
classify all orthondrmal and weakly orthonormal algebraic extensions of
the rationale Q
.
An element of a field is said to be totally positive if and only if
it is positive under every ordering of the field.
Under this definition,
all non-zero elements of a not formally real field are totally positive.
It can be shown that a non zero element of a field is totally positive if
and only if it is a sum of squares.
Lemma 4:
If a field
(<(_ is weakly orthonormal then \<£. admits at most
one ordering.
Proof:
Let ex' be an arbitrary element of
either the equation
X
-R zL
^
then for some integer
or
^
- 19
has a solution in
Theorem Jii
.
Thus, either <=<
or - of is a sum of squares.
The weakly orthonormal algebraic extensions of the rational a
are precisely those algebraic extensions of the rationale
which have at most one ordering.
Proof;
By Lemma 4, the only algebraic extensions of the rationale.which '
are weakly orthonormal are those which have at most one ordering.
K. be an algebraic extension of the rationale
at most one ordering.
First suppose that
a result of Siegel (
"3 » Satz I, p.-^259) there are
(S ^
such that z E 2 ^ ^
—
— 6- «
simple four-dimensional space over ^
^
and
^
=
t
Ue^W
K
such that K_
Then
F
, 14. C «=.4- .
3^
^
admits
is not formally real.
Let (F^
.
Let
By
^ ecV 2. } <=<*3
be an arbitrary semihas an orthogonal basis
Now the field
is not formally real, thus
every five-dimensional quadratic form in five variables with coefficients
has a
in
non trivial zero in . Q C ^ ii -• -•; ^
by the results of the Hasse-Minkowski theory.
1^
.
i.
^ sf
has a non trivial zero in Q (cy^ ^
Thus, either the vector space (Fj
an isotropic vector in which case (F,
so also a vector of length -'I-.
we may choose In -=
orthonormal.
Thus the quadratic form
and so in
'has a vector of length - <L or
contains a. hyperbolic plane and
Therefore, since (F,
was arbitrary,
in Theorem K of Chapter I to show that
Now suppose that
fC is weakly
{<C admits exactly one ordering.
Let (Fj4)
again be an arbitrary semisimple four-dimensional vector space with ortho­
gonal basis £&ij)
in ^
i.
^ 4nd
, 1.-4: t
4
.
Now every element
is either a sum of squares or its negative is the sum of squares.
20
-
.
K n
Thus, for each L , there are
or —
' •
p
such that either p.
Ve now observe that either the quadratic form
or the quadratic form
ia indefinite at every
archimedian completion of the global field Q iCcYlii* *
f **l %
l )'
^ 9'
and so at least one of the forms has a non trivial, zero in.
( cY 1 li * * 'j0 ^. Pzi) and. !so in
V^,
.
Hence, by the same argument as for
the not formally real case, every semisimple four-dimensional space over
f<L contains a vector of length
in Theorem K
Corollary J3:
to show that
tL
1C.
or —
<L ,
so we can take
Yl
4r
is weakly orthonormal.
The orthonormal algebraic extension fields of the rationals
are precisely the not formally real ones.
Proof;
Use Theorem 5» Theorem G and Corollary I.
Remark:
Let K. be a formally real algebraic extension of the rational
numbers with exactly one ordering.
simple vector space over \C «
so E
Let (,Ej^
By Theorem 5,
admits an orthogonal decomposition,
has an orthonormal basis and
Le a
E
-dimensional semi­
l<- is weakly orthonormal,
C
"=
E.-" » where E v*
has a negative orthonormal basis#
Ve
shall show that Sylvester's theorem holds in this case (See L.83 j
Theorem 2.2, p.S2$.).
E-.-
Indeed, if ET
<S E
as the sum
then every vector E+ 1
E:
P
-=
If the dimension of
*t- GL
E.
admits of two decompositions
with
<3^
and
G
€ T- ~~
were less than the dimension of
p
then there
is at least one non zero vector X e in E A E
positive and negative length, a contradiction.
EX
and
EL
can be written
are unique ■( t
;
•
would therefore be of
Thus, the dimensions of
and E l are not) and so we see that the two
- 21 (cardinal) numbers are a complete set of invariants for,non-degenerate
C^o -dimensional forms over \<L .
Remark;
Ve end this chapter by indicating an example of an extension
field of the Gaussian numbers
Consider the polynomial ring
QC-J-I?')
which is not orthonormal.
CvFT?^ L X 0 ,X 1 ( X
Q
* * * *
• 3
It is easy to show by induction that the quadratic form
-
i t < Y K-''-v
has no non trivial zero for any IAZ-CL in Q
Thus, the vector space_(_E ^
Q
('F2 ') C x o , X
CG ^
^
—
in
IV
*• I
C X j
r'
‘ * ’ "I
over the quotient field of
with orthogonal basis
X
fore, the quotient field of .C p C
orthonormal.
C X o ,Vs
^\ i
has no orthonormal basis.
U L X c>4
‘3 is not
There­
CHAPTER IT
HERMITIAH FORMS
Let K
be a commutative field and let <T~
such that cr* o cr =
Let
be the fixed field of cr
If (E" ^vV) is a semisimple
H
(K
then
E
be an automorphism of JC
in K- »
- dimensional vector space with
has an orthogonal basis
CfS d
^ > -v. •
Now for
any X £ E .» by the definition of a Hermitian form, H(1XjX) “ H (.X,Ys) where
<5^" ( HCvkj
E-
V — H L X jx \
•
Thus, the lengths of all basis vectors 6 *^ of
are in the. fixed field K d- .
By restricting the scalars of
the fixed K c-T we may induce a vector space (EL^H)
that E
is a
-vector space with X ^ EE
with X. <5 K i5-
> and the form U -
bilinear form. Transition from
If
E
to
in the natural way
if and only if X is a symmetric non-degenerate
to K <2 > G
gives us E
back again.
X cs- .
K d- is an orthonormal field then
basis and so ( E j I-Vv)
E
f jr > VHs) has an orthonormal
has an orthonormal basis;
weakly orthonormal, then ( E j V-V) has a Jr ± , -
likewise, if K ts- is
IL
orthogonal basis.
We
have proved
Theorem 4:
Let
K- be a commutative field and <r*
such that
l)
2)
.
Then
if the fixed field of 6 "
ef*
n
'
0 <£"-=- HL
an automorphism of
in K
is orthonormal then
is orthonormal.
if the fixed field of <3"" in K- is weakly orthonormal,
then kT" is weakly orthonormal.
We now turn our attention to the problem of determining the orthonormal and weakly orthonormal automorphisms of an algebraic number field.
- 23 In Example 2 below we shall show that an algebraic number field may admit
an orthonormal automorphism' and an automorphism whose square is the
identity automorphism, but which is not even weakly orthonormal.
This
example indicates that not only the field, but the individual automorphisms
of the field must be taken into consideration.
Thus, it seems more pro­
fitable to give a classification criteria which can be applied to special
cases.
Theorem 5 and Corollary 9 below will give necessary and sufficient
conditions for classifying an automorphism over an algebraic number field.
Lemma 5.:
K- be an algebraic field and cf
Let
such that <5“ o <5“ - "IL
in
IC then either l)
exists O,
Proof:
If K tf-
G.
.
V L tfT ^
.
over
Theorem 5.:
or 2) K tf-
K c5-
I —
and Q
C ICrf-
Hence, C. VL \
But both ^ X b V ©
K- and there
KL-
„
VL-
with
and so
and <5"(O') -V-©
are in ItC 3..
Thus,
is the minimal monic polynomial of
K tf-O
Let tC be an algebraic number field and
of
K
such that <
5"O &
if and only if either
^
^=. K —
VLtf- and-
positive element of
Proof:
ICrf. is the fixed field of <5"
" Kl
V ‘S'i X 4- <£"(&)' &
X a -
If
such that K L tf- (
KL then
C VLL- Q * 3
0
«
an automorphism of f<.
=L
^
.
an automorphism
Then 0 " ■is orthonormal
K tf- is orthonormal o r ■there is
is a square root of a totally
K 16- «
Ve first prove the sufficiency.
If
VL
is orthonormal then we
- 24 are done "by Theorem 4.
of any element „o*«
v/here „ X
<5~
So suppose ..
K.
t,
»
is of the form H
^
Then the norm
^ .°v ■”
. W e have by hypothesis that ^
^ c -Ksr
is totally positive in
*
Now let .(,Ej
vector space with
be an arbitrary semisimple |V^ -dimensional
R
(.\<L ,
.
We want to show that
C E ) VA^
has
an orthonormal basis and we shall do this by the methods used
by Fischer and Gross in the proof of Theorem I ( 2 » pp. 287- 288).
.shall show first that
hyperbolic planes.
of
E" _.
?
(E
for .L t: p
±,
Let
.
Suppose that
.
X.
C
the assertion.
N o w ^.<2^ —
Suppose, l\£n \\
is the smallest integer such that
p
^ CT
X
where
-v- <s,
O .
<2 ^
Since C vrt €=
E ?
and
such that
generate a hyperbolic plane in
and
is semisimple,
O
El ^
.
»
Hence ,
«e^
and
and this may be used as the
desired,
-V - i -
Q
Suppose. U €.
dimensional subspace of
3—
Ep
H
assume without loss of generality that <2 ^ - 0
then there is 'y G: E ^
Case 2)
such that
then we have proved
;
Rp
EL
If we can find a hyperbolic plane,
such that
■y
be a fixed but arbitrary basis
S are hyperbolic planes, and <2 ^
where
• L.
p
;y» 4 - E Z ,
Case l)
has an orthogonal decomposition into
j
Suppose that we have found
i
We
b !<.52vXi
scalars of
Let
K-X €
generated by
C. ^
be the one­
, then
*
f~"
to
E „
.
where
p"
is semisimple.
we define the vector space
R
Restricting the
with symmetric
- 25 f orm.
- U
•
Using Theorem 2 of Chapter III we find five
mutually orthogonal}linearly independent vectors ^
1
P
(and thus in
We multiply
p
by
)
11^ tIl-
such that K
.
Since
^
^ ^^ ^
<s^ (
^
= W
in
-'it* O .
is the additive inverse
of a totally positive element, the quadratic form
+-
has a non trivial zero in
call this zero,
*-■<&:$ xg-
VC ^
by the Hasse-Minkowski theory.
Cvl or,
linearly independent, then
^
%
•
-=
X <£ P
such that
hyperbolic plane in
length;
CaJ
and
EL*"
basis with
.
^
l/V <£: P
But
^
has an orthonormal basis and
E
-
.
EE.
Now
X. and
generate a
\\ UU \V
^ ^
EL
Then ^
E
^ ] is a basis of
the scalars of
K ^ r then
(E“
is a vector space over
^
VAI
__
■ .
-
Hence,
t- — E
L
E.
has a negative orthonormal
.
and
>
H
—
X
» each L ,
We restrict ourselves to
^ EL' W =
We note that
Let
Then define ^ ^
is totally positive by hypothesis.
EL
„
EL ^
has an orthonormal basis.
each L ..
form
is semisimple so there
We shall now complete the proof of the
IE
H
•
such that
be the negative orthonormal basis of
y\
(—
By Case I and Case 2 we have shown that
sufficiency by showing that
where
-f- -=V5. ^
Q ^ will generate the hyperbolic plane
cA w Ia
are
and a hyperbolic plane contains a vector of any
thus, there is
which is desired.
where
p
H (Xj
j ^ 3j
<=r'^
is a non zero isotropic vector in F
is
Since
We shall
2"^A ^
j
#
with non-degenerate bilinear
V-\
is symmetric.
Now. the
- 26 length of every non zero vector in
(,k: ,
^
is totally positive;
thus , by the Hass e-Minkowski theory, every five-dimensional semisimple
E.
subspace of '
has a vector of length
of E
to find a subspace
1%
where the
E,p
Suppose we have been able
» such that
Efp -
Or
are four-dimensional vector spaces with orthonormal
bases and such that
integer such that
, say
»
^
for
SL ^
is not in
C
ET ^
«
;
Let
V\
be the smallest
if we can find a space
of dimension four with an orthonormal basis such that
T
A
-
(g:
^
P
i
-
(3-^ Jk.
^ Gr. ©
orthonormal basis.
How let
then we have shown that
(^Vj
Since
U
basis
then
E p1" —
has an
EL ^
[\
is totally positive, then
}U ^ ^
IcL4^
c^> is a symmetric bilinear
X1 be
^
with
UU
=•
W
V
has an orthogonal
•
the one-dimensional subspace of
Vy *
.
Now W
E
A_
is semi simple and the
lengths, of all non zero vectors in V v/ 'are totally positive.
Thus, we
can find three mutually orthogonal, linearly independent vectors V
V
in
WV^W.
^
W
» V.
2*
such that
AV U ^ U ,.
W ~
U U^W
,
Thus, the subspace G -
V z ^ V 3 v V^. 1
has an orthonormal basis.
E_
.
be an arbitrary four-dimensional vector space over
LA ^
We let
CL.
Assume, without loss of generality, that
with an orthonormal basis (the form
form).
T- -
N V 3U - W U 5 U
generated by
is orthogonally is omorphic to
Hence, G -
has an orthonormal basis.
is the
Hence,
,
so <5
(_V
Grpjy.-^. desired and so
Fr ~~ has an orthonormal basis.
This concludes the proof of the sufficiency.
i
»
- 27 It remains to prove the necessity*
normal .
If, <S“ *=
orthonormal.
^
If
Now let £
basis and
where
VC
be a
with the form
then by definition
then
K s-('I’o1^ •=- VC.
So suppose <5“
there exist. cV iu ^
rr — •'i all
Hence,
^ <* •
Cv^
^
"2:C
•
and
\fa!
.,IaA
<A
K s^
5
each
€2
.
.
^ supplied
^ is an orthogonal
Since CS-
is orthonormal
in VC for some
\
O
is
l<_— \<
, such that
But. H
for some.. XT™
^
Sndy by Lemma 5,
-vector space with basis
14 C ClCjts-). such that
VA CG.^
\<L ~ VC
VCtf. <== \<C
C K G VCs-.
is ortho-
= TT^-
l u J" 0,
Thus, we have that
•2.
u y . •=• o
5
< '
Now. if
is not positive under some ordering of
Kg-
the above
equation can have no solution. (We can assume that K c- is formally real,
for if not, then
C\
Ck.totally positive in
is totally positive to start with.)
» which was to be shown.
So we must have
This concludes the
proof of Theorem 5«
Remark;
We have seen by the proof of Theorem 5 that'the property *there
are X 1 ^ * "■ *
,
in the field
K-
such that
—
— CL-
in the case of classifying orthonormal algebraic number fields is equiva­
lent in the classification of arbitrary automorphisms <£~ ( GTo ^
to the property 'there exist, y
^X ^
^
in the field such that
We see by the proof of Theorem 5.that*we may state the following
corollary;
^
*
- 28 Corollary
Let
K-
be an algebraic number field and <s~
morphism of
CT
such that G** <9 S'
K.
an auto­
-=
Then
.
is weakly orthonormal if and only if either \<L
weakly orthonormal or there is
K
—
<$-
is
and
is a square root of a totally positive element in K cj- „
We shall end this paper with several examples illustrating the
above classification criteria.
Example I:
Consider the field,.
morphisms of
ant^
K-<5- "
which sends
- Q
The four auto­
( 3 lla
< 3 ( 3" ^
( vTS? ^
(
}
,
into
-
and has fixed field
^
<5^ which sends
<p
By Theorem 5» ^
• But, since 2 » "I
*and sTT*
into -r ' S
Note that
^"■2. * ^ 3 an^
K-
vpT1 into - 'fT*. and has fixed field
which sends
KLiTt =
( ' T T ) .
are
- lL ^
K
(3
into ~4 7 ' ;
•=: <5”^ ^ <5^ ■=. <S^ o
~
°
is not even weakly orthonormal.
and
are totally positive elements, by Theorem 5
are all orthonormal.
essentially only three distinct
Thus, over this field, there are
-dimensional semisimple vector space
with non symmetric Hermitian forms. (Each of these admit of an orthonormal
basis.)
Example 2i
VL
are
Consider
\C ~
(.vfl? ) (,\[- X*')
.
The automorphisms of
- 29
^ K-
K
±
5 . 1^ O .
xfi? -*» - >/a?
vTT
— (S) (vl-i1
5
— > _ 'i-1.
•= .
^ a - : ^
-
s% ; O 1 - >
--TZ > ^
Note that:
'F£
© <5^_ —
^
© <5^ —
C
vj
(
J zJ
'i
- Q c n n
,
» <5^
— 'Ily^
-
<S^ © 6 ^
c5^ is an orthonormal'field by Theorem J0.
<3^_ is orthonormal by either Theorem 4 or Theorem 5»
<5"g is not even
weakly orthonormal by Theorem 5 and Corollary 9„
CT^ is orthonormal by Theorem 5®
Example
Q ('Pl? ^
The field
has one weakly orthonormal automorphism
which is not orthonormal and one orthonormal automorphism ( S L vc.') »
Example 4:
Let \<
be a splitting field of
Galois group G(K-I q n) is of order 6.
that
31. ^ is orthonormal,
roots of
— 3
in
— 3
are not of order 2 in the Galois group so there
Q
(3
phisms are weakly orthonormal®
The three automorphisms which
.
q
^
and have fixed
Hence, by Theorem 4, these automor­
The square roots which give \<i xwhen
adjoined to the fixed fields are all of the form
orthonormal
The
The two automorphisms which move all three
move exactly two conjugates are of order 2 in G ( K \
the fixed field;
.
Corollary 8 of Chapter III gives
are no Hermitian forms connected to them®
fields isomorphic to
over G)
3 vP
with %
in
thus, by Theorem 5, these three automorphisms are not
*
LITERATURE -CITED
L. E. Dickson,■Theory of Numbers Vola II 1 Chelsea, New York, (1952)
2. R. Fischer and H. Gross. 'Quadratic Forms and Linear Topologies
II', Math. Ann. 1£2, (1964), 296-325»
2» Jacobson, Lectures in Abstract Algebra Vol. II 1 Van Nostrand,
Prince ton, (195 3)«
[J+\ N 1 Jacobson, Lectures in Abstract Algebra Vol. III. Van Nostrand,
Princeton,
[51
I. Kaplansky, 'Forms in Infinite Dimensional Spaces'. Annais Acad,
Eras. Ci XXII. (1950),
;
"
S, Lang, Algebra. Addison-Wesley, Reading, (1965).
Lfl
0. T 1 O'Meara, Introduction to Quadratic Forms. Academic Press, New
York, (1963).
LCl
L. J. Savage, 'The Application of Vectoral Methods to Metric
Geometry', Duke Math. J 1 13, (1946), 521-528.
[9I
G* L. Siegel, 'Darstellung total Posxtiver Zahlen durch Quadrate',
Math, Zeitschr1 11. (l92l), 246-275, .
Lid
L. van der Waerden, Algebra I, Springer-Verlag, .Berlin-GottigenHeidelberg, (1959)» ■
3 1762 I66I0995 6
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