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A Regular Quadratic Form Over a Field K of Characteristic Revista de la 115 Unión Matemática AqeDtiDa VQlumen 84, 1988. ROUND QUADRATIC FORMS ENZO R. GENTILE A regular quadratic form � over a field K of characteristic ,¡. 2 is called l'ound if either it is hyperbolic or i t is aniso­ tropic, satisfying the following similarity conditions : for any K r€presented by the isometry < > holds . x E �, x � � �, We shall study in this Note, round forms over a linked field v mainly with u - invariant u(K) � 4. If dim � = 2 .l, v � 2, l odd , this study was made by M.Marshall [M] . We here complete it to forms of dimensions 2l, l odd . We rely on results in [M] . 1. PREL IMINARI ES . K will denote a field of characteristic ,¡. 2. Quadratic forms over K will be regular (i .e.non-degenerate) and written in diagonal form cal "", an>, a i E K. Jí � and I{J are quadratic forms , � 1 I{J deno tes orthogonal sum and � .I{J, tensor product o If � is a quadratic form, with D(\Ú) we denote the set of all elements of K represented by and � DN) := D(�)\{O} . For a , ..., a K , the n - fold Pfister form l n E < 1 , a > . < 1 , a > ... < l , a > will be deno ted by «a , a , ..., a ». l 2 n l 2 n A non- empty �ubset T of K will be call ed a pl'eol'del'ing if i t is closed respect to sums and products , Le. if T+T e T -and" --­ T.T T. e 116 A ( quadratic) forrn � over K will be called round if either , i) � is hyperbolic or ii) � is anisotropic and for all x E D(�), x O, <x> .� � , i.e. the sirnilarity factors of � coincide f � wi th D ( �) . d A field K is called Unk e d (or, a Un ke d fi el ) if the classes of quaternion algebras over K forrn a subgróup in the Brauer group of K. We shall use results on linked fields contained in [G 1 ] or [G2 ]' The u-invariant of a field K is by definition : u(K) = rnax{dim q} where q runs over the anisotropic torsion forrns over K. If K is a linked field then it is well-known that u(K) {O, 1 4 8 } E , 2 , � . With W(K) we shall deno te the witt ring of K, consisting of the witt classes of all quadratic forms . Next, we recall sorne basic results that will be needed in this paper o Both are from [M] . We give proof of Proposition 1 a which avo ids the use of the Hasse-invariant . 1.1. PROPOSITION ( [M] , Prop.l.1 ( ii)) . be round form L e t � a of dimension 2 • .e.3 .e. odd. Then D(�) e D« l,det�» . Proof. Let a E D(.) , so <a> .� ��, and hence <l , - a > .� = O in the ring of K. This means that � E Ann « l ,-a» =: annihilator ideal in W(K) of <1 , -a> . Now i t is well known (see [EL] , Cor . 2.3) that we have an isometry � ","{Jl l • • • l{J.e. ' where (Ji <c 1 > .<1 ,-b 1 > , bi E D« l ,- a» . Therefore -det � = b b D« l ,-a» . 1 .e. E Thus <1 , - a> = <-det�,det�.a> or <l , det�> = <a, de��.a> , and so a E D« 1 , det�» . 117 1.2 PROPOSITION ([ML Prop . 2.7 ). Let K be a linked fi eld with u (K) Let VI be a round" form over K of dimension 2v.e. odd. ..;; 4 . ¡ , .e. Then i) If v = 2 , there exists a unique v-fo ld Pfister form VIO de­ fined over K suah that VI «de tVl> Vl ) (VI "'" < 1 > 1 Vl )' "" .e. x e Ó o Ó ii) if v ';;¡;. 3, there exists a un�que v-fo ld Pfister form VIO and a unique universal 2-fold Pfister form p defined over K suah that VI "".e. « detVl» VIl) where VI is defined by x e I 2H . ( H denotes a hyperbolia plane) . VIO e p "" VI e 2. ROUND FORMS OVER LINKED FIELDS . 2.1. PROPOSITION. Let K be a linked field and VI a round form of dimens ion 2.e. .e. > 1, odd . Then , i) VI "" 1 �Xi>�i 1 < 1,detVl> with �i, 2-fo ld Pfister forms, xi E K; ii) q 1 <x . >� . is round fo rm and D(VI) D(q) ; := a J. J. iii) D(VI) = D« 1 , detVl» = D(q) ; Iv) D(VI) is a preordering . Proof. If .e. = 1 then VI = <1_ , a>, and so we can assume .e. > 1. i) B eing K a linked field we can wr ite (see [GI l) .�. , VI "" 1 <y. > 1 <a, b> . J. J. with �. , 2-fold Pfister forms . J. Clearly, det VI = a . b. I f we multiply VI by <a> we get VI <x . >�. 1 < 1 , detVl>. "" <a>VI "'" 1 J. J. D(VI) ii) From Prop.1.1 we have e D« 1 ,detVl» and then from i) it is clear that D (VI) = D« 1 , detVl» . iii) is consequenc e o f i) and ii). Iv) Let x,y E D(VI) . Then E D( l <xi>�i) and y E D« l,detVl» x 118 and so x+y e De.) . Por the product x.y, it is clear that x.y E D(.) . 2.2. u(K) PROPOSITION . Let K be a �inked fie�d with � 4 and �et .be a round fOÍ'm o [ dimension U., .e. = 2 k+l , k > O. 1) Assume k odd. Then, :there exista a unique 2-fo�d Pfister fo rm « a,b» suah that i) • "" k«a, b » 1 <l ,det.> ii) D« l ,det.» = D« <a,b» ) iii) D« <a,b» ) is a preordering . r 2 ) Assume k = 2 .h, r � 1, h odd. Then, there exists a unique (r+ l)-fo �d Pfister form .0 and a unique universa� 2-fo�d Pfister form p suah that i) • "" h.1 1 < 1 ,det.> where .1 is a round form defined by .0P'P "" . 1 1 2H . ii) D(.1 ) is a preordering. iii) D(.l ) = D« l ,det.» . Proof. 1) By using Prop . 2.1 i) we can wr ite • "" 1 < X.>'P . 1 <l, detlP> . - 1 1 with : 'P 1- , 2-fold Pfister fcirm, and q :=1 <X - >'P - a round form of dimens ion 4k, k odd . 1 1 By applying [M) , 2.7 ( i) it fol1ows the existence of a unique 2-fold Pfister form « a,b» such that 'P O k.« a,b» . q "" Therefore k. ,det.> . • "" o 1 <1 If k = 1, then D« <a,b ») = D« l,det.» and we know that D« l,det.» is a preorder ing . Assume then k>1. That k. «a,b» .is a round form impiies , by using �) 1.7, that D« <a,b») is a preordering . So D« <a,b» ) =D(k« a,b ») = D« l,det.» . 1 119. " 2) Assume k= 2 r . h, r :>1, h odd .With the no tation of Prop . 2 .1 i) we have that q = 1 <x i >ep iis a round form of dimension 2r+l .h, h odd . It follows from �] 2 . 7 , the existence of a unique v +l­ fold Pfister form �O and a unique universal 2 - fold Pfister form p such that where ep I is defined by epO 1 p "" ep1 1 2H. The refore � "" hep l l <1 ,det�> . By P�op.2.1 we have fhat D« l ,det�» . If h = j, then ep l is round and D(ep l ) is a preordering . If h > 1, then" by [M] , 1.7 i t follows that D(epl) is a preorder ing and ep I is round . 2.3. PROPOSITIQN . Let K be any fieLd and Let � be an a.nisotr-o­ pie for-m over- K, and �l a r-ound fopm. Then if � can be wr-itten as � "" k'�1 1 <1 ,det�> with k E N, k odd, and D(�l) = D« l,det�» a pr-eor-der-ing, then � is a r-ound fo r-m. Pr-oof. Since D(�l) is preorder ing K, it fol lows from a � [MI , 1.7 that k�l' is a round form if k> 1 . lf k=l , the same is clearly true . Le"t x E D(�), wr ite x = X l + X 2 ' X l � D(k. l ), X 2 E D« l,det�» . Then, X E D« l,det�» = D(� ) Therefore l + �2 l ' <X l + X2 > ' �1 "" � l' (X l + X2 ) · k�1 "" k�l and 120 <X l x >.< 1,det�> < l,det�> . + 2 � Consequently <X>.� <X X > . � � , l + 2 � and � is round . REMARK. 2.4. If K is a global field and � is an anisotropic round form over O [HJ]). K then dim .K (mod and det I/J = 1 (see In fact, == 4) since D« l,det�» is a preordering, it follows that <l ,det�> represents all sum of squares . Therefore,. for every discrete prime p in K we have that < l,detl/J>p is universal in the compl� of K. Now , according to [OM] , 63:15 (ii) if is a �ion Kp � two -dimensional anisotropic form over a local field K and if 2. � represents 1, then D(�) is a subgroup of K of index The ­ refore our form <l ,det�> is isotropic for all but finite num a ber of spots (the real ones) . Equivalently - det� is a square in a1l, but a finite number of K ' [O M] , 65: 1 5, we conclude p By that - det� is a square in K, i .
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