ALMOST ISOTROPIC QUADRATIC FORMS

ALEXANDER PRESTEL AND ROGER WARE

In this note we give a valuation theoretic characterization of formally real fieldsfor which every totally indefinite quadratic form is almost isotropic.

The notations and terminology will primarily follow [4]. Thus by a field F we willmean one of characteristic different from 2 and we denote by (al,...,an) the

n

diagonalized quadratic form £ a , ^ 2 , tf.ei7, of dimension n. Every quadratic

form considered here will be regular, that is all at # 0. The quadratic form <al s . . . , an}will be called almost isotropic if for every diagonalization (bu ...,&„> = <tf1} -..,«„>there is an m e N such that mQ>xy 1 <62, •••> bn} is isotropic in the usual sense, while<#!, ...,«„> is called weakly isotropic if m<fll9 ...,«„> is isotropic for some m ^ 1.Of course, an isotropic form is almost isotropic and an almost isotropic form isweakly isotropic. The form <,Y, 1, — 2x} over Q(x) provides an example of a weaklyisotropic form which is not almost isotropic, since for the diagonalization<1, x, — 2*> there is no meN such that w<l> _L <;c, — 2x> is isotropic.

The notation of weak isotropy has been introduced in [5]. A form <al5 ...,#„>is called totally indefinite if for every ordering P of F there exist indices i, j with a , e Pand ttj e - P. The fields for which every totally indefinite quadratic form is weaklyisotropic have been characterized by Satz 3.1 in [5]. The class of these fields turnedout to equal the class of SAP-fields (see [3; Theorem C]). The notion of almostisotropy is used implicitly in [8]. There, the form <«j, ...,#„> is called effectivelydiagonalizable if it is isometric to a form (bx,...,&„> satisfying b^ P =>bi+leP forall 1 < / < n and all orderings P of F. The field F is said to satisfy ED if every formover F is effectively diagonalizable. Theorem 2.4 of [8] can be stated as:

THEOREM 1. Afield F satisfies ED if and only if every totally indefinite quadraticform is almost isotropic.

Proof By Theorem 2.4 of [8], F satisfies ED if and only if every form prepresenting 1 over all real closures of F represents a totally positive element of F.

Assume F satisfies ED. If (Jbu ...,£„> is totally indefinite, then

_ / **p~\ Tr-~

represents 1 over every real closure and hence represents a totally positive elementte F. Since / is a sum of squares, m^b^ _L (b2,..., £„> is isotropic for some m.

To prove the converse, let p represent 1 over all real closures of F. Then< — 1 > ± pis totally indefinite so there exists m i n N such that /w< —1> J_ p is isotropic. Thus prepresents a totally positive element of F.

The aim of this note is to prove the following:

Received 20 March, 1978; revised 12 September, 1978.The second author gratefully acknowledges support from NSF Grant MCS 76-06581.

[J. LONDON MATH. SOC. (2), 19 (1979), 241-244]

242 ALEXANDER PRESTEL AND ROGER WARE

CHARACTERIZATION THEOREM. A field F satisfies ED if and only if for everyvaluation v: F -> G with formally real residue class field Fv we have \G/2G\ < 2 andFu euclidean in case \G/2G\ = 2. (Recall that a field is euclidean if it is uniquely orderedand every positive element is a square).

This theorem follows from Theorem 1 and the equivalence of (4) and (6) inTheorem 2 (below). Let us point out that for non formally real fields the equivalenceis trivial. A valuation v : F -*• G will be called a real place if the residue class field Fv

is formally real.

THEOREM 2. For a field F the following properties are equivalent.

(1) The pythagorean closure of F is SAP.

(2) F(V0 is SAP for all totally positive t in F.

(3) F is SAP and every torsion form {a, by over F represents a totally positive t in F.

(4) Every totally indefinite quadratic form over F is almost isotropic over F.

(5) For all a, b in F, the form (\,a,b, —ab} is almost isotropic.

(6) For every real place v : F -*• G we have

(a) |G/2G| < 2

(b) if \G/2G\ = 2 then Fv is euclidean.

Proof (1) => (2). This follows from Satz 2.8 in [5].(2) => (3). Since F = F(V1), F is SAP. Let <a, by be a torsion form over

F. Then m(a, by is hyperbolic for some m ^ 1. So there exists a totallypositive t in F with a+tb = 0. Since Fx = F(V0 is SAP there exists ain Ft such that

a e P if and only if y/teP and —aeP, for all orderings P of Ft. (*)

Since <a, by = <1, —1> over Fu Scharlau's Norm Principle (see [4; Corollary 4.5,p. 207]) implies that N(a) • {a, by £ <<z, by over F, where JV = NFl/F is the norm.Thus <fl, by represents aN(a). Let P be an ordering of F and let P^ be theextension of P to Ft containing y/t. If' denotes the conjugation of Ft over F, thenPi is the second extension of P to Ft. Since -Jt = y/t'eP^, yjt$ P'. To showthat aN(a) lies in Pu consider first the case -as P. Then by (*), ae Px and <x£ P / .Hence - a ' e Pj so that -JV(a) 6 P = P± n F. Thus aN(<x) e P. Now assume thata e P. Then a £ Px and a £ P / . Hence - a e P x and - a ' e P i so that aN(a) e P.This proves that {a, by represents the totally positive element aN(cc).

(3) => (4). Multiplying {a, by in (3) by a suitable scalar, we see that <a, byactually represents every coset of F modulo the totally positive elements.

Now let p be totally indefinite over F and <o1} . . . , O a diagonalization of p.Since F is SAP there are sums of squares ty,...,tn in F, not all zero, such thatax tx + ...+an tn = 0. Let 1 ^ k < n be the smallest integer such that there existssuch a representation with tk+l,..., tn in F2. Then

for some totally positive / / , . . . , f*_i', ffc in F. Hence the binary form

ALMOST ISOTROPIC QUADRATIC FORMS 243

is torsion and therefore represents some — ax t with t totally positive. Thus thereexist c, d in F with

which contradicts the minimality of k, unless k — 1. Hence p is almost isotropic.

(4) => (5) is trivial.

(5) => (6). By (5) all forms <1, a, b, — ab} are, in particular, weakly isotropic.Hence by Satz 3.1 and 2.2 of [5], we have \G/2G\ ^ 2 and Fv is uniquely ordered incase \G/2G\ = 2. It remains to show that Fv is actually euclidean when \G/2G\ = 2.Let a e Fv be positive. Then a = J^a2 for some at in F. Choose b in F with v(b) $ 2G.Because (\,a, b, — ab} is almost isotropic over F, there exists m in N such thatm(\, a) 1 (b, —ab} is isotropic. Now a is a sum of squares in the formally real fieldFv so the residue class form < 1, —a> must be isotropic over Fv. Thus a is a square in

iv(6) => (1). Since direct limits of SAP-fields are again SAP-fields, it suffices to

show thatif K satisfies (6) then K(ja) satisfies (6). (**)

Let yx : K^ -*• Gx be a real place of K± = K{Jd). Since (6) holds for K, (a) and (b)hold for v = V]|K and G = v(K). Since \GJG\ < [1^ : K] < oo, it follows that|Gi/2G,| = |G/2G| < 2 and that Kv is a euclidean field in case \GXI2GX\ = 2. But(K1)Vl is formally real and [(K1)Vi : Kv] ^ 2 so if Kv is euclidean we must have(Ki)Vl = Kv, completing the proof.

We now obtain a sequence of corollaries.

COROLLARY 1. / / all the real places of F have 2-divisible value groups then Fsatisfies ED. This in particular holds if

(a) F has only archimedean orderings, or

(b) F is an algebraic (not necessarily finite) extension of a uniquely orderedfield.

An extension K of F is called a 2-extension of F if K is contained in the quadraticclosure of F. For example, the pythagorean closure of F is a 2-extension of F.

COROLLARY 2 (Going up). Let F satisfy ED. Then every 2-extension of F alsosatisfies ED. (In particular, the pythagorean closure of F satisfies ED.)

Proof. This follows from (* *) in the proof of Theorem 2 and the fact that directlimits of ED-fields are again ED-fields.

COROLLARY 3 (Going down). Let K/F be a finite extension such that all orderings ofF extend to K. (This holds, for example, if [K : F] is odd or if K = F(y/t) with ttotally positive.) If K satisfies ED then so does F.

Proof. We use (6) of Theorem 2. Let v : F -> G be a real place of F. Since allorderings extend, v extends to a real place w : K -> Gx. Then |G/2G| = \G1/2G1\ ^ 2.

244 ALMOST ISOTROPIC QUADRATIC FORMS

If \G/2G\ = 2 then \GJ2Gil = 2soKw is euclidean. Since [Kw : Fv] is finite, Fu is alsoeuclidean.

Like SAP, ED does not go down a quadratic extension in general. A counter-example is provided by Remark 3.4 of [7]. Also, if Fo is the euclidean closureof Q then the formal power series field F0((x)) satisfies ED but possesses a galoisextension of degree 3 which is not ED.

Let us now turn to the connection of ED with the finiteness of the Hasse numberu. As in [7], let u(F) be the smallest number n < oo such that every totally indefinitequadratic form of dimension ^ n+\ is isotropic over F. Obviously, u(F) < ooimplies ED for F. Conversely, it was shown in [8; Theorem 3.1], that for ED fieldsF with only finitely many orderings, u(F) < oo implies u(F) < oo, where u(F) is theusual K-invariant (see [2]). From the formulation of ED in Theorem 1 one mightexpect that w(F) is finite if not only F but also all algebraic extensions of F satisfy ED(as, for example, is the case for algebraic number fields). In this case we may call Fhereditarily ED. However, from Brocker's result [1] that there are uniquely orderedfields with infinite Pythagoras number, it follows that this is not true. For byCorollary 3, F would be hereditarily ED while the finiteness of u(F) would implythe finiteness of the Pythagoras number P(F). In [7] it is shown that even theadditional assumption P(F) < oo does not imply ii(F) < oo. In Theorem 3.1 of[7] a uniquely ordered subfield F of U is constructed satisfying P(F) = 2 andu(F) = oo.

In a forthcoming paper of Richard Elman and the first author Theorem 3.1 of [8]will be strengthened to:

u(F) < oo and F satisfies ED if and only ifu(F) < oo.

From this result, Theorem 2, Corollary 3, and Theorem 7.4 of [2] we obtain

COROLLARY 4. Suppose K = F(y/t) w'ltn * totally positive. Then ii(F) < oo ifand only ifu(K) < co.

References

1. L. Brocker, " t)ber die Pythagoraszahl eines Korpers " Arch, der Math., 31 (1978), 133-136.2. R. Elman and T. Y. Lam, "Quadratic forms under algebraic extensions", Math. Ann., 219

(1976), 21-42.3. R. Elman, T. Y. Lam and A. Prestel, " On some Hasse Principles over formally real fields ",

Math. Z., 134 (1973), 291-301.4. T. Y. Lam, The algebraic theory of quadratic forms (W. A. Benjamin, Reading, Massachusetts,

1973).5. A. Prestel, " Quatratische Semi-Ordnungen und quadratische Formen ", Math. Z., 133 (1973),

319-342.6. A. Prestel, Lectures on formally real fields (Monografias de Matematica 22, Instituto de Matematica,

Pura e Aplicada, Rio de Janeiro, 1975.)7. A. Prestel, " Remarks on the Pythagoras and Hasse number of real fields ", / . reine angew. Math.

303/304 (1978), 284-294.8. R. Ware, " Hasse Principles and the w-invariant over formally real fields ", Nagoya Math. J., 61

(1976), 117-125.

Fachbereich Mathematik, Department of Mathematics,Universitat Konstanz, Pennsylvania State University,

Postfach 7733, University Park, PA 16802,7750 Konstanz, West Germany. U.S.A.