Specialization of quadratic and symmetric

A C T A A R I T H M E T I C A X X I V (1973)

ndes, M a t h .

University,

Specialization of quadratic and symmetric

bilinear forms, and a norm theorem

by

M A N F B E D K N E B U S C H (Saarbrücken)

(338) Dedicated to Carl Ludwig Siegel on hü 75 birthday

Introduction. I n t h e first p a r t of this paper ( § l - § 3 ) we s t u d y t h e specialization of a symmetric bilinear o r quadratic f o r m over a field K w i t h respect t o a place X: K->Lvoo, p r o v i d e d the f o r m has " g o o d reduc- t i o n " . W e have t o distinguish between s y m m e t r i c b i l i n e a r a n d quadratic forms since we do not exclude fields of characteristic 2 . A t y p i c a l result obtained b y this theory is the f o l l o w i n g : W e denote a s y m m e t r i c bilinear f o r m b y the corresponding s y m m e t r i c m a t r i x of its coefficients. L e t k(t) be the field of r a t i o n a l functions i n independent variables t ^x , t ^r over a field k. Consider symmetric bilinear forms (/#(*))> over k(t) whose coefficients f^ij{t)¹ g^kl(t) are polynomials. Assum e t h a t the f o r m (g^ki(t)) i s represented b y (fq(t)). Assume further t h a t c is a n r - t u p e l i n Jf such t h a t the f o r m (fij(c)) over k is n o n singular. I f charfc # 2 t h e following holds t r u e :

(i) I f also (g^ki(c)) is n o n singular, t h e n this f o r m i s represented b y [faie)) over k (see § 2 ) .

(ii) I f (g^ki(t)) is a diagonal m a t r i x w i t h m rows a n d columns a n d if c i s a n o n singular zero of each p o l y n o m i a l g^t), t h e n the f o r m (/#(£)) has W i t t index > m/2 if m is even a n d > ( m + l ) / 2 i f m is o d d (see§3).

The assertion (i) m a y be considered as a generalization of the principle of s u b s t i t u t i o n of Cassels a n d Pfister ([15], p . 3 6 5 ; [10], p . 20). A t t h e end of Section 3 (Proposition 3.6) we shall also generalize t h e subform theorem of Cassels a n d Pfister ([15], p . 366; [10], p . 20).

U s i n g t h e result quoted above a n d a similar result f o r charfc = 2 we p r o v e i n the last section § 4 a theorem about the p o l y n o m i a l s i n k[t]

w h i c h c a n occur as norms of s i m i l a r i t y over Jc(t) f o r a f i x e d symmetric bilinear f o r m defined over k. Special cases of this n o r m theorem have been used i n a crucial w a y b y A r a s o n a n d Pfister i n [1] a n d b y E l m a n a n d L a m i n [5].

I n general our results about quadratic forms are m u c h less complete t h a n those about bilinear forms.

A l t h o u g h t h e language of forms is quite n a t u r a l t o describe t h e m a i n results of this paper, we use i n the b o d y of the paper the geometric language of quadratic a n d bilinear spaces, since t h e geometric language seems t o be more suitable to understand t h e proofs.

The theory developed here w i l l be applied i n a subsequent paper about t h e behavior of quadratic forms i n transcendental field exten- sions [9].

§ 1. Preliminaries about bilinear and quadratic spaces. W e recall some standard notations a n d w e l l - k n o w n facts about symmetric bilinear a n d quadratic forms over a (not necessarily noetherian) local r i n g A.

F o r proofs of statements given here w i t h o u t further reference a n d moreover for t h e basic theory over a r b i t r a r y c o m m u t a t i v e rings t h e reader m a y consult Chapter V of [3], [13], [7] a n d § 1 of [8]. I n t h e present paper essentially o n l y t h e case t h a t A is a field or a v a l u a t i o n r i n g w i l l p l a y a role.

A free (symmetric) bilinear module (EjB) over A is a finitely gen- erated free ^.-module E equipped w i t h a symmetric bilinear f o r m B:

ExE->A. W e often denote (E, B) b y a symmetric m a t r i x (a^{j) w i t h a^{j

= B(x^{, Xj) fo r some basis x¹⁹..., xⁿ of E over A. W e say t h a t (E, B) — or B — is non singulary or t h a t (E, B) is a bilinear space, i f det(a^) lies i n t h e u n i t group A* of A^r i.e. i f x\-^B{—, x) is a bijection f r o m E t o the d u a l module Hom^(_E, A). A free quadratic module (E, q) over A is a finitely generated free ^.-module E equipped w i t h a quadratic f o r m g, i.e. w i t h a m a p p i n g q: E-+A such t h a t q(cx) = c²q(x) a n d B(x,y):

— q(oc + y) — q(x) — q(y) is bilinear i n x a n d for c i n JL, x a n d y i n E.

W e say t h a t {E^r q) — or q — is non singular or t h a t (E, q) is a quadratic space if the associated bilinear f o r m B is n o n singular. A quadratic module (E, q) w i l l often be denoted b y a s y m m e t r i c m a t r i x [a#] i n square bracket w i t h a^u = q(Xi)j a^{J = B(x^iy Xj) if i # j , for some basis x^x, xⁿ of E.

If 2 is a u n i t i n A there is n o essential difference between quadratic a n d bilinear modules, since then a n y bilinear f o r m B corresponds t o a unique quadratic f o r m q(x) = \B(x, x).

F o r a free quadratic module we always denote t h e quadratic f o r m b y t h e letter q a n d t h e associated bilinear f o r m b y B as far as n o con- fusion is possible, a n d we often write E instead of (JE7, q). S i m i l a r l y we denote the bilinear f o r m of a free bilinear module usually b y the letter B, a n d we often write E instead of (E, B). I f we use the w o r d " s p a c e " w i t h o u t further specification we regard bilinear a n d quadratic spaces a t t h e same time. T h e r a n k of a free finitely generated J.-module V w i l l be denoted b y d i m V.

L e t

<p(l)=l (p*(E) t h f o r m w h no 4, § 3 clear w h L e t module ¹ other sul If A is c consistin for EjT- W e submodn case and E is anitt over A ] t r o p i c q A qi isotropic to t h e o:

tains a d to the o

E v e position (*) w i t h E⁰ r a t i c ca>

(1.0) for quae thermor the n u n ; determr write t case the still uni<

space o:

t = i n d .

complete ribe the eometric language it paper i exten-

"e recall bilinear r i n g A.

iioreover ler may- it paper

i l l p l a y ely gen- 'orm B:

w i t h a^{? 7} F. B) - .%) lies om E to

~er A is f o r m q, B(x, y):

I! i n E.

uadratic module bracket

a O f E.

itic a n d unique ic f o r m no con- a r l y we otter B, w i t h o u t

ie same lenoted

L e t q>: A-+A' be a h o m o m o r p h i s m between (local) rings (of course

^(1) = i ) . F o r any free bilinear or quadratic module E we denote b y .(p*(E) the JZ-module E^^A' equipped w i t h the bilinear resp. quadratic form w h i c h is deduced f r o m the f o r m on E b y base extension ([4], § 1 no 4, § 3 no 4). W e often write E®^AA' or E®A! instead of <p*(E) if i t is elear w h i c h m a p q> is considered.

L e t E be a free quadratic or bilinear module over A. W e call a sub- module V of the JL-module E a direct submodule if E = V@W w i t h some other submodule W ( © means the module s u m , w i t h o u t regarding forms).

If J . is a v a l u a t i o n r i n g then for a n y submodule V of E the module V¹ consisting of a l l x i n E such that B(V, x) = 0 is a direct submodule, for E/Y¹ is torsion free a n d finitely generated a n d hence free.

W e call the bilinear or quadratic module E isotropic if E has a direct submodule V # 0 w h i c h is totally isotropic, i.e. q(V) = 0 i n the quadratic case a n d B(VxV) = 0 i n the bilinear case. If E is not isotropic, we say E is anisotropic. Notice tha t if A is a field a n y anisotropic bilinear module over A must be a space, b u t t h a t i n case char J . = 2 there exist aniso- tropic quadratic modules w h i c h are not spaces.

A quadratic space E over A is called hyperbolic, if E contains a t o t a l l y isotropic direct submodule V such t h a t V^L = V. T h e n E is isomorphic to the orthogonal sum txH of t = .J-dimJE? copies of the hyperbolic plane E =|J Jj.- S i m i l a r l y a bilinear space E is called metabolic, if J0 tains a direct submodule V = T^{r J}-. A metabolic bilinear space is isomorphic to the orthogonal sum of spaces

con-

w i t h some a i n A.

E v e r y quadratic resp. bilinear space E has a n orthogonal decom- position

(*) E •= E⁰±2I

w i t h E⁰ anisotropic a n d 31 hyperbolic resp. metabolic. N o w i n the q u a d - ratic case W i t t ' s cancellation law is true, since J . is l o c a l [6], i.e.

(1.0) E\ AG ^ I¹* ±G =>F^X g± F²

for quadratic spaces F^±,F²,G over A ("^" means " i s o m o r p h i c " ) . F u r - thermore M ^ txH w i t h some > 0. Thus i n the decomposition (*) the number t = J-dimitf a n d up to isomorphism the space E⁰ are uniquely determined b y E. W e call t t h e index of E a n d i<7⁰ a kernel space of JE? a n d write t = indJS, J?⁰ = Ker(.2?). If is a/iWd of char 2 t h e n i n the bilinear ease the cancellation l a w fails, but the space E⁰ i n (*) is u p to isomorphism still u n i q u e l y determined b y E ([7], § 8.2, [11]). W e again call J^o a kernel space of E a n d t: = i d i m i l / the index of E a n d w r i t e E^Q = K e r ( i ? ) , t = inAE.

W e call t w o bilinear resp. quadratic spaces E a n d F over A equivalent, a n d write• E ~ F, i f there exist metabolic resp. h y p e r b o l i c spaces J f a n d N such t h a t E± M ^ F±N. I f well-defined k e r n e l spaces exist, this means Ker(E) ^ Ker(jF). T h e equivalence class of a space E w i l l be denoted b y {E}. F o r a n y space E we denote b y —E t h e m o d u l e E equipped w i t h t h e f o r m —B resp. —g, where B resp. q denotes t h e f o r m of t h e original space. T h e space E J_( —E) is always metabolic resp. hyper- bolic. T h u s t h e equivalence classes of bilinear or quadratic spaces f o r m a n abelian group under t h e a d d i t i o n {E} + {F} = {E±F}, a n d t h e i n - verse of a class {E} is {—E}. T h i s group is called the Witt group W(A) of bilinear spaces resp. t h e W i t t group Wq{A) of quadratic spaces over A.

I n fact, W(A) i s even a c o m m u t a t i v e r i n g under t h e m u l t i p l i c a t i o n {E}. {F} == {E<g)F}. H e r e E<g> F denotes the tensor p r o d u c t of the A - m o d u l e s E a n d F equipped w i t h t h e tensor produc t of t h e bilinea r forms of E a n d F ([4], § 1 n o 9). (Furthermore Wq(A) is a Tf (A)-module; w e shall not need this fact.) C l e a r l y a r i n g h o m o m o r p h i s m y: A-+A' (with <p(l) = 1 of course) induces homomorphisms W(<p): W(A)-+W{A') a n d Wq(<p):

Wq(A)->Wq(A') w h i c h m a p th e class {E} of a space E to t h e class {^(2*7)}.

W e n o w give a description of t h e r i n g W(A) b y generators a n d relations. A n y bilinear space E over A w h i c h contains a n element x w i t h B{x,x) i n A * has a n orthogonal basis, i.e.

E ^(aj±...±(a¹⁷) ,

w i t h some a^ti n A * . A s usual we denote the r i g h t h a n d side also b y (a¹,...

aⁿ). N o t i c e t h a t i f 2 is a u n i t of A every bilinear Äpace E # 0 con- tains some # w i t h J3(#, a?) i n A * . A n y w a y f o r a n a r b i t r a r y l o c a l r i n g A the r i n g W(A) is a d d i t i v e l y generated b y t h e classes {(a)} of spaces of r a n k one. W e write {a} instead of {(a)}. L e t G denote t h e group A*[A*² of square classes <a> = a A *². W e have a r i n g h o m o m o r p h i s m O f r o m the group r i n g Z[G] onto W(A) mapping- (a) t o {a}. L e t m denote t h e m a x i m a l ideal of A . T h e following w e l l k n o w n theorem w i l l be used i n this paper o n l y f o r tn = 0 .

T H E O R E M 1.1 ([19], S a t z 7, [7], § 5 , [8], § 1 ) . Assume A/m contains more than two elements. Then the kernel of 0 is additively generated by the elements <<*>•+<•—<*> and the elements <a¹y + (a²}--<b¹y--<b²y such that (a^x, a²) 9*'(&!, 6²)> which is the case if and only if (a^xa^) = <&i&2>

6^X = c²a¹ + d*a² with c and d in A.

W e close this section w i t h some remarks o n q u a d r a t i c modules.

T h e following generalization of W i t t ' s cancellation theorem is a n imme- diate consequence of Satz 0.1 i n [6].

P R O P O S I T I O N 1.2. Let M and N be free quadratic modules over a local ring A and let G be a quadratic space over A. If G±N represents G±M then N represents M.

F o r consisting say t h a t (in differ<

the quas t h u s E n has a d«

w i t h so:

the n u i n determin oiE. A n ;

V h a s i n

®r@vr a space, (e.g. [7]:

due to t

§ 2 . w i t h K i t i o n rin£

of h L E 3 over o s case eve If c a n d the Pre ->Wg(ii T h i s is (In [13 Prüfer : M¹ ~ 2 case 21 W i w i t h re of f u l l : B y L e n m i n e d s depend even t l .

equivalent, spaces i f aces exist, ace E w i l l module E the f o r m :sp. hyper- paces f o r m ad t h e i n -

oup W(A) 'es over A.

implication A-modules Orms of E y^f we shall h<p(l) = 1 i d Wq(<p):

™{<P*(E)}.

gators a n d ent x w i t h

v #-0 con- al r i n g A

spaces of up A * / A *² a <P f r o m lenote t h e je used i n n contains ated by the such that

^ib²} and modules, a n i m m e - ver a local nts G±M

F o r a n y quadratic module E over a field K we c a l l t h e submodule consisting of a l l x i n E w i t h 2? (a?, JE7) = 0 t h e quasilinear part of JE7. W e say that E is non degenerate if t h e quasilinear p a r t of 25 is anisotropic (in difference to the terminology i n [6]). N o t i c e t h a t i n the case c h a r K ^ 2 the quasilinear part of a n o n degenerate module E m u s t be zero a n d thus E must be a space. A n y n o n degenerate quadratic module E over K has a decomposition

B a r x [ i o ] ^{I En}

w i t h some r > 0 a n d E⁰ anisotropic (cf. [2], p . 160). B y P r o p o s i t i o n 1.2 the number r a n d u p to isomorphy the quadratic m o d u l e E^Q are u n i q u e l y determined b y E ([2]). W e call r t h e index of E a n d E⁰ a kernel module of E. A n y m a x i m a l t o t a l l y isotropic submodule V of E has r a n k r. Indeed, 7 has intersection zero w i t h the quasilinear p a r t E of E. T h u s E = B©

© V@W w i t h some other module W. T h e module U = V@W m u s t be a space. Thus V is contained i n a hyperbolic space M cz U of r a n k 2 d i m F (e.g. [7], Satz 3.2.1). W e have E = J f X J i "¹ a n d J f^x m u s t be anisotropic due to the m a x i m a l i t y of V.

§ 2 . Good reduction of spaces. W e consider a f i x e d place A: K->Luco w i t h X a n d i fields of a r b i t r a r y characteristic. W e denote b y o t h e v a l u a - t i o n r i n g of /, b y m the m a x i m a l ideal of o a n d b y ja t h e restriction o->2/

of A.

L E M M A 2.1. Assume JI¹ and 3I² are (quadratic or bilinear) spaces over o such that ^ J/²<g>⁰jEL. Then M^x ~ M². In the quadratic case even 31\ ^ 31².

If c h a r i # 2 there is of course no d i s t i n c t i o n between the quadratic a n d the bilinear ease.

P r o o f o f L e m m a 2.1. Since o is a Prüfer r i n g t h e maps Wq(o)->

-+Wq(K) a n d W(o)-*W{K) induced b y th e i n c l u s i o n o->K are injective.

This is p r o v e d i n [7], § 1 1 , or [13], p . 93, i n t h e bilinear case.

(In [13] only D e d e k i n d rings are considered, b u t t h e proof holds for Prüfer rings.) T h e quadratic case c a n be settled i n t h e same w a y . T h u s 31¹ ~ 31². Since 31 ^± a n d 31² have the same r a n k we o b t a i n i n the quadratic case .JI^X ^ 3I² (see § 1). q.e.d.

W e say that a quadratic or bilinear space E over K has good reduction w i t h respect to A, if E contains a quadratic resp. bilinear space over o of full r a n k , i n other words, if E ^ 3I®⁰K w i t h some apace 31 over o.

B y L e m m a 2.1 the space p*(3I) is u p to W^tt-equivalence uniquely deter- m i n e d b y E. W e denote the class i n W(L) b y A* {JE?}. I t clearly depends o n l y of the class {E} i n W(K). I n the q u a d r a t i c case b y L e m m a 2.1 even the space /.i*(3I) over L is u p t o i s o m o r p h i s m u n i q u e l y determined

b y E, a n d w i l l be denoted b y A* (JE). W e call A*(2?) the reduction or spe- cialization of E w i t h respect to A. Assume now that Charly = 2 a n d E is bilinear. W e say t h a t E has very good reduction, if E contains a bilinear space i f over o of full r a n k such that the space J/*/mi/ = i f ®o/m over o/m is anisotropic. T h e n for a n y other space i f ' over o of f u l l r a n k con- tained i n E we o b t a i n f r o m i f ' / m i f ' ~ i f / m i / " that i / / m i f is isomorphic to a kernel space of i f ' / m i f ' (see § 1) a n d thus i f ' / m i f " ' ^ i f / m i f , since the ranks are equal. T h u s also i f ' / m i f ' is anisotropic a n d p+(M) ^ /i^m(M'), W e again call ⁱu * ( i f ) the reduction or specialization A* (2?) of 2?.

The later E x a m p l e 2.6 (i) shows t h a t i n the bilinear case w i t h c t i a r i

= 2 "good r e d u c t i o n " is not enough to ensure the uniqueness of A* (2?).

P R O P O S I T I O N 2.2. Let E = F ±G be an orthogonal decomposition of a space E over K.

(i) If E and F have good reduction, then also G has good reduction and X*{E) = / * { i ? H / * { £ } .

In the quadratic case even

h{E) ^^(F)U*(G).

(ii) Assume E is bilinear and c h a r i = 2 . If E has very good reduc- tion and F has good reduction, then F and G both have very good reduction and again

A*(2?) ^ A*(F)_LA*(G). .

E e m a r k . W e shall see i n § 3 (Proposition 3.2) t h a t i n assertion (ii) the assumption that F has good reduction can be dropped.

P r o o f . W e chose a decomposition G = G⁰±G¹ w i t h G⁰ anisotropic a n d G^x hyperbolic resp. metabolic. I t is easy t o f i n d a space B^x over o of f u l l r a n k i n G^x. I t remains to f i n d such a space i n G⁰. Clearly G⁰ is a kernel space of E±(—F). W e chose spaces i f , JN⁷ over o of f u l l r a n k i n E a n d F. W e further chose a decomposition i f JL( — N) ^ B⁰±S into a n anisotropic space B⁰ a n d a hyperbolic resp. metabolic space 8. The space 22⁰®⁰2T is again anisotropic (see [7], §11.1), hence

B⁰®⁰E ^KeT(E±(-F)) ^G⁰,

a n d G ^B®⁰E w i t h B: = B⁰±B¹. W e see that G has good reduction , a n d obtain f r o m E ^ (N±B) ®⁰E, tha t /*{£} = A* {F} + A* {G} a n d i n the quadratic case A* (2?) ^ X*(F) J_A*(G).

Assume n o w t h a t c h a r i = 2 a n d E is a bilinear space w i t h v e r y good reduction. T h e n NlmN'±B/mB is anisotropic. T h u s b o t h summands are anisotropic a n d assertion (ii) follows, q.e.d.

If i f a n d J\^T are quadratic or bilinear free modules over a local r i n g A, we say t h a t i f is represented b y N a n d write i f < JS^r, if ^ c o n t a i n s

a direct is a spac*

module 1 to the 1 of space;

C o i u K and n

(i) I (ii) - is anisotr

P r o coincides a regulai d u c t i o n « t i o n r i m Clearly the quot hypothe, w i t h the canonica quireme:

W e 2.2 w i t h E g±F„

E e t h a t JST/

" s t a b l y (b) field -BL jective.

..: A s Co:

r of van be an r

(i) and if is repn

(ii)

(/«(*)),

•w or spe~

2 a n d E i bilinear o/m over ank con- omorphic since

fch c h a r X of A*(#).

sition of otion and

)ä reduc- reduction

r t i o n (ii) isotropic

\ oyer o rly ö⁰ is

a l l r a n k _!_# into e 8. T h e

uction, a n d i n ith v e r y mmands

>eal r i n g contains

a direct submodule M' (isomorphism respecting t h e forms). I f J f is a space t h i s implies X M ±T w i t h a suitable quadratic resp. bilinear module T. O f course i f X a n d i f are b o t h spaces also T is a space. U p to t h e last p a r t of this section we shall only deal w i t h representations of spaces b y spaces.

C O R O L L A R Y 2 . 3 . Assume A is a regular local ring with quotient field K and maximal ideal 9ft and that JH and J V are spaces over A.

(i) In the quadratic case M®^AK < X®^AK implies M/3JIM < N/WIN.

(ii) If 2 € 301 the same holds true in the bilinear case, if in addition N/WIN is anisotropic.

P r o o f . I t is easy to construct a place A: K-+A/Soluoo w h i c h o n A coincides w i t h t h e evident m a p A-+A/3JI. I n fact, l e t t^lf ...,t^r denote a regular system of parameters of A. W e show t h e existence of A b y i n - duction o n r. I f r = 1 take the canonical place associated w i t h the v a l u a - t i o n r i n g A. Assume n o w r > 1 a n d l e t p denote t h e p r i m e ideal At^r. Clearly A^p is a v a l u a t i o n r i n g a n d L: = A^v/pA^p m a y b e regarded as the quotient field of the r i n g A/p, w h i c h is again regular. B y i n d u c t i o n hypothesis there is a place a : L-*AfiBtuoo w h i c h coincides o n Afp w i t h t h e evident m a p from Afp to A/SR. L e t ß: K-^Luoo denote t h e canonical place associated w i t h A^p. T h e place A = aoß fullfills our r e - quirements.

W e now obtain the assertions of Corollary 2 . 3 a p p l y i n g P r o p o s i t i o n 2.2 w i t h E: = X®^AK, F: = M®^AK a n d 0 a space over K such t h a t E g*F±G. q.e.d.

E e m a r k s 2A. (a) I f i n p a r t (ii) of Corollary 2 . 3 w e d o n o t assume that is anisotropic, then i t still c a n b e shown t h a t J f / 2 R J f i s

" s t a b l y represented" b y N/WIN, i.e. J f / 9 K J f _ L # i s represented b y XjyRN±S for some space S over AjWl.

(b) I t is u n k n o w n whether for a regular l o c a l r i n g A w i t h quotient, field K t h e canonical maps W(A)->W(K) a n d Wq(A)->Wq(K) are i n - fective. Corollary 2 . 3 gives a small h i n t t h a t this m i g h t b e true.

A s a special case of Corollary 2 . 3 we o b t a i n

C O R O L L A R Y 2 . 5 (Principle of substitution). Let (/#(*)) and (ffia(t))i<it^ftem be symmetric matrices of polynomials in an arbitrary number r of variables t. = (t^t,.... t^r) over an arbitrary field Jc. Let further c = {c^x,..., c^r) be an r-tuple with coordinates c^t in a field extension L of k.

(i) If the quadratic modules [g^ki(c)] and [/#(<?)] over L are nonsingular and if the module [g^ki(t)] over k(t) is represented by •[/#(<)], then [g^kl(c)]

is represented by [/^(c)].

(ii) The analogous statement holds for the bilinear modules (g^ki(t))y (fn(t))> (9ki(e))i {fij(c)) tf ^{w e} assume in the case charfc = 2 in addition that (fij(c)) is anisotropic.

4 P r o o f . [g^kl(t)] resp. (g^ki(t)) is a fortiori represented b y [/#($)] resp. (/#(*))

over £(J). T h u s i t suffices t o consider the case L = 7;. 2sow a p p l y Corollary 2.3 w i t h A = ftp]^p, where p denotes the ideal of ftp] generated b y — c^{1 9} . . . , ^ — c^r. q.e.d.

B e n i a r k . Corollary 2.5 (ii) is i n the case m = 1 a n d a l l f^i}{t) con- stant t h e well k n o w n principle of substitution of Cassels a n d Pfister ([15], p . 365, [10], p . 20). I n this case no additional assumption is needed if char ft = 2. I n fact, we m a y assume again that L = ft. L e t E denote the space (/#) over ft. W e consider a decomposition

* - * 4 *

S) ^ f r i) H I J)

w i t h E⁰ anisotropic a n d m i n i m a l r. T h e n the subspace

i = E⁰±(a¹, «r)

is anisotropic. (#ⁿ(/)) is already represented b y E<g>k(t), a n d o u r Corollary 2.5 shows t h a t (gu(c)) is represented b y E, hence b y E.

E X A M P L E S 2.6. (i) Assume ft is a field of Charakteristik 2 w h i c h is not perfect, a n d l e t a b e a n element of ft w h i c h is n o t a square. T h e n w i t h one variable t t h e spaces (1,'1 + ai²) a n d [a. a(l + at²)) over ft(t) are isomorphic. S u b s t i t u t i n g £ = 0 we o b t a i n t h e spaces ( 1 , 1 ) a n d (a, a) over ft, w h i c h are n o t isomorphic. (1, l + at²) represents (a) over k(t) b u t (1,1) does n o t represent (a) over ft. This shows that even for m = 1 a n a d d i t i o n a l assumption is needed i n Corollary 2.5 (ii) i f char ft = 2 a n d the f^{j(t) are n o t constant.

(ii) F o r m = 2 a n d char ft = 2 already a n additional assumption is needed i f a l l f^{j are constant. F o r example w i t h t h e element a f r o m above t h e spaces ( 1 , 1 , a) a n d (a + t% a + t², a) over k(t) are isomorphic (see [11], Theorem 3, or [7], Satz 8.3.1). Thus (1,1, a) represents t h e anisotropic space (a, a + t²) over k(t). B u t ( 1 , 1 . a) does n o t represent (a, a) over ft. Indeed, otherwise ( 1 , 1 , a) w o u l d be isomorphic t o (a, a, a) w h i c h is absurd, since (a, a, a) does n o t represent (1).

W e n o w want t o prove a generalization of P r o p o s i t i o n 2.2 i n t h e quadratic case. A s above l e t /: K->Luoo denote a fixed place w i t h v a l u a t i o n r i n g o. W e t a c i t l y assume u p t o t h e end of this section t h a t c h a r i = 2, since otherwise L e m m a 2.8 a n d Proposition 2.9 below are already proved.

L E M M A 2.7. Let N be a free quadratic module over o such that NJmN is non degenerate. Then N is maximal among the lattices Is' over o in N®K with q{N') c o.

P r o o f

w i t h N^x a decomposii p . 259). S i : i n the subi i n E such

Xi i n N^{i since N^x i?

implies x²

L E M M

MlmM an Then Nim sotropiCj t P r o o o/m. W e f to show 1

_ L ( - J f ) .

for some prove o u i now.

W e r bolic p l a n of t h e o- B{x, N) = sees, that 2.7. Thus a v e r y c l of N is n more ox-

.(b) "

of L e i n i i i over o an*

a n d thus 1.2 w e ( to show the proof tions of

MM*))

oroUary

It) con- Pfister needed denote

>rollary

liich is . T h e n '(t) are i (a, a) er k(t)

m = 1 -Jc =2 t i o n is i. f r o m lorphic

ts the

>resent

•'h.a, a) in the*

> w i t h a t h a t w are X/nuV

P r o o f . There exists some decomposition

T / T T U V = N^t±N²

w i t h JN^ a space over o/m a n d N ² the quasilinear p a r t of NjmN. This decomposition can be lifted to a decomposition N = J ^ i J J V ^ (^.g. [6], p. 259). Since ÄF² is anisotropic i t is easily seen t h a t N² is the set of a l l z i n the submodule y²®K of E w i t h g ( z ) € 0 . INOW assume t h a t a? is a vector i n E such tiiat:g(jV + o#) is still contained i n o. W e have x=x^x + x2 w i t h

i n jSf^{®E. Clearly 5(a?, A⁷^) =\B(a?^{1 ?} J ^ ) <= o. T h i s implies a^cS^, since JVY is a space. Thus X + ox = ^ + 0 0 ^ . I n p a r t i c u l a r q(x²)eo, w h i c h implies ^²€-Ä^{T 2}, as states above. This completes the proof, q.e.d.

L E M M A 2.8. Let 31 and N be free quadratic modules over 0 such that i f / m J f and 'NjvxN are non degenerate. Assume that N®K represents If<g>2L Then J^/miV* represents 3Ifm3I. If 31 is a space over 0 or if N/mN is ani- sotropic, then even N represents 31.

P r o o f , (a) W e shortly write M for M/mM, N for F/mN, a n d Jc for o/m. W e first consider the special case t h a t 31 is a space over 0 a n d have to show I f < J. B y P r o p o s i t i o n 1.2 i t suffices to p r o v e 31 J_( — M) < JV_L

_ L ( - J f ) . JSOW

3I±(-3I)

a r x [ j J]

for s o m e r > 0. Thus we see again b y P r o p o s i t i o n 1.2 t h a t i t suffices to p r o v e our assertion i n the special case 31 ^ ^ jj, w h i c h we consider now.

W e regard N as a lattice of E: = N<g>K. Since Ü7 represents a hyper- bolic plane, there exists some x i n N w i t h q (x) = 0 a n d ox a direct s u m m a n d of the o-module X The ideal B(x,N) of 0 is f i n i t e l y generated, thus B(x, J¥) = ao w i t h some a == 0 i n 0 . A s s u m e a em. T h e n one i m m e d i a t e l y sees, t h a t g takes on'N + a^xo o n l y values i n 0 . T h i s contradicts L e m m a 2.7. Thus aeo* a n d there exists some y i n N w i t h B(x, y) = 1 . (This is a v e r y classical argument, see e.g. [14], p. 235). T h e submodule ox + oy of N. is n o n singular a n d i n p a r t i c u l a r a direct s u m m a n d of N. F u r t h e r - more ox + oy is isotropic a n d t h u s hyperbolic.

(b) W e now consider the general case. A s e x p l a i n e d i n the proof of L e m m a 2.7, we have a decomposition 21 = 3I¹)L_M2 w i t h M^x a space over 0 a n d B(II² x 31²) <= nt. W e k n o w b y p a r t (a) of t h e proof t h a t M¹ < JV, a n d thus JV ^ 3I¹±F² w i t h some quadratic module J V⁸. B y P r o p o s i t i o n 1.2 we o b t a i n f r o m 31 ®K < N®K t h a t l/²®JBL < N²®K. I t suffices to show t h a t 3I²jm3I2 is represented b y j$^r2lmN2. W e t h u s have reduced the proof to the special case t h a t B (11x31) c m i n a d d i t i o n to the assump- tions of the proposition.

W e again regard J as a lattice i n E: = IS⁷® K a n d regard i f as a lattice i n F: = i f ® IT. W e assume w i t h o u t loss of generality t h a t J ⁷ is a submodule of Ü7 over K. N o w the intersection X^t: = 3⁷n J P is con- tained i n i f since i f is the set of a l l z i n F w i t h i n o. B y t h e elemen- t a r y divisor theorem there exists a basis . . . , %^m of J f a n d a basis Vu Vm °* ^1 ^{s u}° k * b^a* 3/i ^ ^at ^ '^w ^ ^a i ^ⁿ o. W e m a y assume t h a t there is some s i n [0, m ] such that a^{ = 1 for 1 < i < $ a n d c^cm for

$ < i < m . I f s = m , t h e n J T2 = i f a n d thus i f < J\^T. C e r t a i n l y s = m if N is anisotropic. Since n o w we assume s < m. L e t V denote the image of t h e direct s u m m a n d N^r of A⁷ i n N. Since J B ( i f x i f ) c m we h a v e 5 ( 1 ! x f j) c tn a n d thus B(YxY) = 0. L e t a^{ denote t h e image of q(%i) i n k. T h e n

(*) l ^ W i . . . i W , a n d

F a W l . . . i W l ( ^ ^ ) x [0].

L e t further V⁰ denote t h e intersection of Y w i t h t h e quasilinear p a r t B of N. Since V⁰ is a n anisotropic submodule of Y. clearly F⁰ is represented b y i.e.

(**)

w i t h J = d i m Y⁰ a n d suitable elements c^{1 ?} i n fc*. ( B e a d Y⁰ for the r i g h t h a n d side i f t = *.) Thus we have a decomposition V = V⁰±U where U is a submodule of Y w i t h

CT [Ci] 1 . . • 1 [c,-*] _L (m - s) x [0].

(The [c^t] have to be omitted if t = Ä.) N O W we choose a submodule TP of N such t h a t

jsr = B,@u®w = i?±(i7eF).

T h e submodule P : == [/©TT must be a space. L e t u^m_^t denote a basis of U w i t h g(^) = c^t for 1 < i < 5 — a n d q(Ui) = 0 for 5 — J < i

^m — t. Since J3( CT xZ7) == 0, we c a n f i n d elements £^m_* i n P such t h a t J3 ^fZj) = 0 a n d JB (^, %) = for i a n d j i n [1, m — t] (e.g. [7], S a t z 3.2.1). T h u s we finally obtain a decomposition

N ^B±(ku¹ + kz¹)±...±(ku^m^ + kz^m^^t)±Q

w i t h some space Q. N o w J B represents Y⁰ a n d JcUi+kZf represents [ c j for 1 < i < $ — t. F o r s — t < i ^ m — t t h e space fc^ + ft^- is hyperbolic a n d certainly represents [ a< + <] . W e see that JV⁵ indeed represents i f , since b y (*) a n d (**)

M F o l [ o¹] i . . . i [ v^/] l [ a^{s + 1}] l . . . l K ] . q.e.d.

W e sa>

w i t h respec f u l l r a n k \ quadratic i the just p i to isomorp

W e nc Propositioii

P B O P O

nearly goo<

MF) < }.

C O R O . . .

for the qua<i

and J T / a i l :

true if the

§ 3 . S

place /.: Ji case t h a t t crete also [ the argum

T H E O I

with A* {a}

h{a} = 0 P r o o a n d /: K ther assui is infinite element a map .1 fr

= {a} if a u n i t of o.

i n K*. A c that A ru

w i t h (a^lr of o, thi>>

T h e n z =•

such t h a¹ ments u.

5 — Acta Ar

gard 31 as i i t y t h a t F

•:~\F is con- he elemen- ud a basis

sume t h a t i a^{€m for nly s = m

the image i we h a v e

imasre of

ear p a r t R epresented

ad 7⁰ for ' = v0±u

module W

_i denote

P s-^t < i

?^m_t i n P ] (e.g..[7],

V

Quadratic and symmetric bilinear forms, and a norm theorem

W e say that a quadratic module E over K has nearly good reduction w i t h respect to A: A % X U C O , if J ? contains a quadratic o-module M of full r a n k w i t h 31/mil/ non degenerate. W e then denote b y A* (22) the quadratic module ,u+(JI) over L w i t h ,u: o->£ the restriction of A. B y the just proved L e m m a 2.8 the quadratic module /.*(JE7) depends up to isomorphy only on E a n d A.

W e now obtain from L e m m a 2.8 the following generalization of Proposition 2.2 for quadratic modules:

P R O P O S I T I O N 2.9. If E and F are quadratic modules over K with nearly good reduction ivith respect to A: I I—> i u o o , and if F < E, then A*(F)<A*(E).

C O R O L L A R Y 2.10. The assertion (i) of Corollary 2.3 remains true if for the quadratic modules 31 and X occurring there we only assume that 3IjyjtM and X/yjlN are non degenerate. The assertion (i) of Corollary 2.5 remains true if the word "non singular¹ there is replaced by ^unon degenerate".

§ 3 . Subspaces w i t h bad reduction. A s i n § 2 we consider a f i x e d place A : ' Ü L - > i u o o . The following theorem is w e l l - k n o w n i n the special case that the v a l u a t i o n r i n g o of A has r a n k 1, see [7], § 12, a n d for o dis-.

crete also [17] a n d [13]. Chapter I V . § 1. W e shall prove it b y generalizing the argument given i n [13].

T H E O R E M 3.1. There exists a unique additive map W(K)->W(L)

with A* {a} = {/.(a)} for every a in K* such that A(a) ===0, oo⁷ and with A*{a} = 0 for every a in K* such that ?.(ac²) = 0 or oo for every c in K*.

P r o o f . W e may assume w i t h o u t loss of generality t h a t L = o/m and A: J L - K L U O O . i s the canonical place associated w i t h o. W e m a y fur- ther assume m == 0. .since else the theorem is t r i v i a l . T h e n certainly K is infinite a n d we can a p p l y Theorem 1.1 w i t h A = K. The image of a n element a of o i n L w i l l be denoted b y ä. W e have a well defined additive map .1 from the group r i n g Z[G]. G: = K*jK*², to W(L) w i t h /l<a>

= {ä} if a in o*, a n d A (a = 0 if the square class (a} does not c o n t a i n a u n i t of o. Clearly this map .1 vanishes on a l l elements <a> — < — a} w i t h a i n IC A c c o r d i n g to Theorem 1.1 our theorem w i l l be p r o v e d if we show that A vanishes on an a r b i t r a r y element

\sents [Ci]

lyperbolic sents M^f

w i t h (a^ly a.2) ^ ( % , at). If none o f^:t h e square classes contains a u n i t of o, this is evident. Thus we assume without loss of. generality a^o*.

T h e n z = \a^x) y w i t h a n element

y = 1 + <c> — <ft> — \bc^

such that (1, e) ^ (6, be), w h i c h means 6 =u*-+v²c w i t h suitable ele- ments uj v of K. F o r a r b i t r a r y elements a i n o* a n d x i n Z[G] we clearly

5 — Acta Arithmetica XXIV.3

have A((a}x) = {a}A(x). Thus i t suffices t o prove A(y) = 0. W e assume that b o t h u a n d v are # 0 , since otherwise already y = Ö.

W e first consider the case that c lies i n o * . T h e n Mv) =(l-r{c})A(l-<b)).

W e have n o t h i n g t o prove if .{c} = {—1}. Thus we assume i n a d d i t i o n {c} # { — 1 } / w h i c h means that t h e space (1, c) over L is anisotropic.

Changing b b y a square we further assume t h a t u a n d v b o t h lie i n o b u t not b o t h i n m. Since (1, c) is anisotropic, we have b = .ü² + cv2 ^ 0 a n d

A(y) = (i + {c})(l-{ü* + c&}) 0 .

W e now consider t h e remaining case that < c does n o t contain a u n i t of o. T h e n u~²v2c is not a unit a n d thus either b = u2(l + d) or b

= t^{, 2}e ( l - f d ) w i t h some d i n m. Hence J (1 — b ) = 0 or = '1 — {c} a n d A(y) 0 i n b o t h subcases. q;e.d.

E e m a r k . F o r a bilinear space E over T T w i t h good reduction the element /*{!?} constructed i n § 2 is the same element as the image of {E} under the map /* constructed now. This follows easily from the fact, that f o r every space 31 over o at least the space J / _ ( l ) has the f o r m ( « ! , . . . , a^f) w i t h a⁽ i n o* (cf. §1).

The map W(K)->W(L) gives some information ' about spaces w i t h good reduction w h i c h contain subspaces w i t h bad reduction, i.e. not good reduction.

P P W O P O S I T I O N 3.2. Let E be a bilinear space over K with good reduction.

(i) Assume e h a r i # 2 . If E, represents a space ..., b^m) ßiich that

?,(b{c2) = 0 or oo for each b{and every c in K*. then /*(£) has. index ^ {m/2}.

(As usual {m/2} denotes the least integer > ml2.)

(ii) If c h a r i = 2 and E has very good reduction, then each subspace of E also has very good reduction.

P r o o f , (i) JE? ^ , . . , - &^m, c² w i t h some c^{ i n K*. Thus h{E} = ?^{{c11 . . . , c „ _^m) } . F r o m the definition-.of /.* i t is clear that the equivalence class of /„*(£) contains a space of r a n k < )t—m. This means t h a t A * ( E ) has a n index ^ {m/2}.

(ii) E must be anisotropic since E has very good reduction. Thus certainly every subspace of E has an orthogonal basis. I f E w o u l d con- t a i n a subspace w i t h b a d reduction, then E would contain a space (b) of r a n k one such that /-(6c²) 0 or oo fcr all c i n K*. B u t then we see again, t h a t A*(J?) is equivalent to a space of lower rank. This contradicts the assumption t h a t /*(£) is anisotropic. X o w the assertion follows f r o m P r o p o s i t i o n 2.2 (ii). q.e.d.

P R O P O S I T I O N 3.3. Let (f^i}(t)\'.be a symmetric (n, nymatrix of polyno- mials fij(t)€k[t^l1 i^r] over an arbitrary field Jc, and g^x(t), g^m(t) be ni

further ^A coordina is a no) i depend h:

(gi(t),~

P r e

r = 1, t:

use of t l X(t^x) = <

T h e n :

for 1 < ; dinates v L e t e' C l a r y 2.3 U s i n g t l ' over /t'(/

w i t h soJ L e t /.: I /*: Wik zero, tht This me

Ass:

B y wha a space ß: ]:((()- of r a n k

E e apply i m

PVJ

near!j/ <>

is a st(fj has an

P I T

non deg(

With 80!

0. W e assume - 0.

ie i n a d d i t i o n is anisotropic, rh lie i n o b u t

- cv- ~ 0 a n d

* not c o n t a i n /(²(1-M) or b -= 1 — {c} a n d reduction t h e the i ma,ere of r'rom the fact, has the f o r m ibout spaces tion, i.e. n o t ood reduction.

b^m) such that idex > {m/2}.

ach subspace in J L \ T h u s

!ear t h a t the . This means action. T h u s

•] w o u l d c o n - i a space (b)

then we see s contradicts

follows f r o m

;x of polyno-

• ^?g^m(t) be m

< 1

further polynomials in l:[t]. Assume c = (c^x, c^r) is an r-tuple with coordinates in a field extension L of k such that det{f^ij(c)) ^ 0, and that c is a non singular zero of each g^pJ i.e. g^p(c)•=. 0, (dg^pjdt^q)(c) # 0 with some q depending on p. Then if the space (/#(*)) ^{o v e r} k(t) represents the space lgi(t)> the space {fij(c)) over L has an index > {m/2}.

P r o o f . A s i n the proof of C o r o l l a r y 2.5 we m a y assume L = k. Ii r = 1. then our proposition follows i m m e d i a t e l y f r o m P r o p o s i t i o n 3.2 b y use of the place A: k(t^l)-*k\jcG over k (i.e. A is t h e i d e n t i t y o n Jc) w i t h }⁹(t^x) = c¹. Assume n o w r > 1. W e first consider the case t h a t & is infinite.

Then there exists a n /-tuple (a^lJ...,a^r) i n Jc^r such t h a t

r

^«^q(0g^pidt^q)(c) ==0

for 1 <J p < m. Thus performing a suitable linear transformation of coor- dinates w i t h coefficients i n k we m a y assume (dg^pjdt^(c) ^ 0 for 1 < p < m.

L e t & denote the (/• — l ) - t u p l e (c², ..., c^r). There exists a place a: k(t) - > h " ( y u x over fc(^) w i t h afa) - e⁴- for 2 ' < * < r. (Cf. the proof of Corol- l a r y . 2.3 w i t h A the local r i n g of k{h)[U. ..., t^r] corresponding to V . ) Using the m a p A*: IFfÄ;(^)) —^TF(Ä*(#^x)) we see t h a t the space (fiⁱ(t¹¹c^r)) over fc^) is equivalent to a space

if/iCivO? g^m(tu <*•')> ^hi(ti)< . w i t h some polynomials Ji^P(ti) a n d w — (If m = omit the Ji^p.) L e t A: kit^-^ku CSJ denote t h e place over k w i t h A(^) = c^x. T h e n using

•A*: T F f k ^ l - ^ - T F ^ ) we see that, since a l l A*{^(^, c')}, 1 < i < m, are zero, the space•(/.,-;(e)|. over 7J is equivalent to a space of r a n k < n — m.

This means that the index of this space is > {m/2}.

Assume n o w that k is finite. L e t u denote a n indeterminate over k.

By..what has been proved the space (fij(c)) is equivalent over k(u) to a space of r a n k < n — m. A p p l y i n g ß * : W{k(u))->W(k) w i t h some i^lace

•ß: h{u)—^kuco over Avwo see that lfij(c)) is over k equivalent to a space of rank //^. q.e.d.

E e in a r k . If eliar// — 2 the proof could have been shortened b y applying first the principle of substitution a n d t h e n P r o p o s i t i o n 3.2 (i).

PROPOSITION 3A. Assume E is a quadratic module over K ivJiicli Jias nearly good reduction ivith respect to A: K->Luoc. Further assume that F is a submodule of K with /.\q(x)) = 0 or co for every x in F. TJien A*(ü7) lias an index dimF.

P r o o f . L e t X denote a module over o of full r a n k i n E w i t h X/mX non degenerate a n d let 31 denote the intersection XnF. T h e n X = 31 ©31' w i t h some .other submodule 31' of X. T h e image M of 31 i n X/mX is

a submodule of t h e same r a n k as J F w i t h q(M)=- 0, since q(oc)€m for all x i n I f . This implies the assertion (see § 1 ) . q.e.d.

PROPOSITION 3.5. Let (/,-;($)) be a symmetric matrix of polynomials fij(t) in k[t] - k[t^x, , . . , tr] and g^t), ...,g^m(t) be m further polynomials

in k[t]. Assume that c = ( c^{1 ?} . . . . c^r) is an r-tuple tvith coordinates c^{ in a field extension L of k, such that the quadratic module [fij(c)] over L is non degenerate, all g^p(c) = G, and the matrix (dg^pjdt^q)(c) has rank m. Assume further that the module over k(i) represents [gi(t)]±... __[g^m(t)].

Then ijie module [fij(c)] over L has index ^ in.

P r o o f . W e m a y assume k = L. F u r t h e r replacing* the variables t^{ b y t^{ — c^{ we assume c =• 0. F i n a l l y subjecting the t^£ t o a suitable linear transformation w i t h coefficients i n k we assume

(dg^ßldt^Q)(0) = <5

for 1 ^ p < m, 1 < q < r. W e consider t h e field I\ = s) w i t h a n i n - determinate s over k(t), a n d the subfield 7;(K) = k(u¹, ...,a^r) of J L w i t h

if. = / . s "¹. L e t / denote the place from K = k(u, s) to Ä-(Y/) over.

w h i c h maps 5 to 0. E e g a r d i n g the module -F = [<M()]

over I i , we shall prove below:

(**) = 0 or cv for a l l # i n JP

Since t h e module [fij(t)] OYQY K represents P . a n d /.(/,) = 0 for 1 < i < v..

it t h e n follows b y P r o p o s i t i o n .3.4, that the module [/y(0)] over k(u) has index > m. B y Corollary 2.10 also the module [/⁰(0)] over A: has index > m.

W e n o w prove •(**)• F o r a n y x ^ 0 i n F the v a l u e g(«r) has the f o r m ZN~^l w i t h

m

X ⁼⁼ h(u, s)², Z.== J T ^ - N , « )²f f f ( « i « , ...,*/>.*),.

Ä(W, *) a n d a^f(w, « ) denoting polynomials i n s] such t h a t 7*(u, «): ^ 0 and n o t a l l a* (M, *) = 0 . W e n o w regard t h e a^f(u, s) as polynomials i n s w i t h coefficients i n k[u]. F o r some 1^0 we m a y write for 1 < f < r

a^{(y, s) = b^{(M)sz + higher terms',

w i t h a t least one b^{(u) # 0 . B y (*) the lowest t e r m of gⁱ(u¹8,'...,'ti:^r8) w i t h respect t o s is u^{s. Thus the lowest t e r m of Z is c(w)$^:

w.

c(u) = y^biWUi,

, 2 / - r l w i t h

provide degree the low O n the X(ZX~'

W(

over 7/(

i n £[/].

space ( extensi*

gular a t i o n 3.c zero th(

ing E ov

w i t h F quely <

P E

above ( 2>artf of

Pi- r a t i c f(

of C a *^: Pfister n o w fc

P r W i t h o i

denotr

w i t h a.

If h(t) linear

q(oc)€xn fo r

polynomials polynomials (mates cⁱ in

•)•] over L is k m. Assume

riables %ⁱ b y table linear

w i t h a n in-- j of K w i t h ) over k(u)

r 1 < i < r, ] over k(u) over k has as the f o r m

•h(ii, s) ^ 0 .omials i n s 'C i < r

'iSj ..., u^rs)

1 w i t h

provided c(u) ^ 0. B u t a glance o n the t e r m of lowest (or highest) t o t a l degree i n the p o l y n o m i a l c(u) makes evident, t h a t indeed c(u) # 0. T h u s the lowest degree w i t u respect t o s occurring i n the p o l y n o m i a l Z is 21+1.

O n t h e other h a n d the lowest degree of N is a n even number. Clearly A(ZN~]) = 0 or = oo. q.e.d.

W e n o w consider the following s i t u a t i o n : (/#(£)) is a bilinear space over k(t) = t^r) w i t h indeterminates t^{ a n d t h e f^{j(t) polynomials i n k[t]. F u r t h e r g(t) is a p o l y n o m i a l i n k[t], w h i c h is represented b y t h e space {fij(t)) over k(t). F i n a l l y c is a n r-tupel w i t h coordinates i n a field extension L of k such t h a t the bilinear module (fq(c)) over L is n o n sin- gular a n d g(c) = 0 . If the zero c of g is n o n singular, t h e n b y P r o p o s i - t i o n 3 . 3 the space (/#(c)) over L must be isotropic. B u t if c is a singular zero then i t m a y v e r y well h a p p e n t h a t (fij(c)) i s anisotropic. The follow- ing Proposition 3 . 6 deals w i t h this case. E e c a l l t h a t a n y bilinear module E over L has a decomposition

E ^ FAX x (0)

w i t h F a n o n singular bilinear module, w h i c h u p t o isomorphism is u n i - quely determined b y E. W e call F the non singular part of E.

P R O P O S I T I O N 3 . 6 . Assume Char ft # 2 and that in the situation described above (fij(c)) is an anisotropic space over L and in particular c is a sin- gular zero of g. Then the space (ftj(c)) over L represents the non singular part of the bilinear module (a^pq), over L with

%^q = i(d²g/dt^pdt^q)(c)..

E e m a r k . I n the special case t h a t a l l /#(£) are constant, g(t) is a quad- ratic form, a n d c = 0 , P r o p o s i t i o n 3 . 6 is the well k n o w n subform theorem of Cassels a n d Pfister ([10], p . 20). I n fact, t h e theorem of Cassels a n d P f ister provides t h e m a i n step i n the proof of P r o p o s i t i o n 3 . 6 , w h i c h now follows.

P r o o f . W e proceed on a similar w a y as i n the proof of P r o p o s i t i o n 3.5.

W i t h o u t loss of generality we assume k = L a n d c = 0. L e t h(t) = Ya^pqt^pt^q

denote t h e Hessian f o r m of g(t), w i t h t h e a^pq f r o m above. W e have g(t) = h(t) + c(t)

w i t h a p o l y n o m i a l c(t) w h i c h o n l y contains monomials of t o t a l degree ^ 3 . If h(t) = 0, there is n o t h i n g t o prove. Thus we assume after a suitable linear transformation of coordinates that for some index m i n [1, r] a l l a^pq w i t h p > m or q > m are zero a n d t h e m a t r i x (a^pq), 1 < p, q < m

has determinant 0. I f m < r we obtain b y t h e principle of substitu- t i o n 2.5, t h a t t h e space (/^(tu t^mJ 0, .... 0)) over A*(*x, t^m) repre- sents g{t^XJ 0 , 0). Thus we m a y assume without loss of gener- a l i t y f r o m t h e beginning, that the space (a^PQ) over k is n o n singular.

W e again consider t h e fields K =k(t,s) = k(u, s) a n d k(n) con- structed i n t h e proof of Proposition 3.5. T h e space

over K represents t h e element

s~~²g(t) = h(u) + sc(u, s)

w i t h c(u, s) a p o l y n o m i a l i n s]. S u b s t i t u t i n g t h e value 0 for s we see b y 2.5, t h a t t h e space (/^(O)) over k(u) represents h(u). S o w t h e subform theorem of Cassels a n d Pfister yields t h a t t h e space (/#(())) over k represents t h e space (a^pq) over k. q.e.d.

§ 4 , The norm theorem. W e consider a f i x e d bilinear or q u a d r a t i c module E over a n a r b i t r a r y field it. W e call a n element a of V a norm of Ü7, i f (a)®E ^ J?, i.e. i f a is the n o r m of a s i m i l a r i t y transformation of JB. (If E is quadratic w i t h associated quadratic f o r m q t h e n (a)<g>E denotes t h e module E w i t h the quadratic form aq.) T h e set of norms

of E i s a group JS^T(E) w h i c h contains a l l squares i n k*.

The m a i n goal of this section is to prove t h e Theorem 4.2 below about the norms of E®k(t) w i t h k(t) = k(t^u t^r) the field of r a t i o n a l functions i n a n a r b i t r a r y number r of variables t^x, ..., t^r over k. F o r a n y p o l y n o m i a l f(t)ek[t] we denote b y /* the coefficient of t h e highest mo- n o m i a l occurring i n f(t) w i t h respect to t h e lexigraphical ordering.

( t ?¹. . . # v > # . . . ^ i f a n d only if the first difference ^ - 6 , ^ 0 is > 0.) W e say t h a t / i s normed i f /* = 1 . Notice t h a t this n o t i o n depends o n the chosen ordering t^t> t²>-.... >t^r of t h e variables. W e further f i x the following notations for this section if r > 1: K denotes the field k(f) w i t h V = (t^2f 2r), d e gx/ denotes the degree of / as a p o l y n o m i a l of K[t^xl

W e shall need t h e following

L E M M A 4.1. Assume that E is an anisotropic bilinear or quadratic module # 0. If a polynomial f{t)ek[t] is a norm of E®k(t) then the highest monomial occurring in f(t) has the form t™¹ ... if^r with even exponents m^{. Furthermore f* is a norm of E.

W e prove t h e l e m m a b y i n d u c t i o n o n r. Assume first r = 1 a n d write t instead of t^x. W e consider the place /.: k(t)->kuoo over k w i t h X(t) = oo. Clearly E®k(t) represents a n element af(t) w i t h a i n k*. I f deg/ w o u l d be odd, t h e n the Propositions 3.2 a n d 3.4 w o u l d i m p l y t h a t E is isotropic. Thus deg/ is a n even number m, a n d hence

E®k(t)& (r^mf(t))®(E®k(t)).

Fürthen the righ

Ass of / as i

is even to h. q.

F o r function of k[t]j\

T i n over k t Then tin

(i) ( « )

= const (in) If ( L e m m a since E group.

Olei.

Cor, Then a is a nor divide f

W e the case T o pro^

over k i to the i

we obta every n

defined