function field over F.) Are such Pfister function fields K always excellent over F? ... The signifcance about the excellence properties of Pfister function fields is as.
Inventione$ matbematicae
Inventiones math. 51, 6 1 - 75 (1979)
9 by Springer-Verlag 1979
Function Fields of Pfister Forms R. Elman 1., T. Y. Lam 2.*, and A. R. Wadsworth 3.** University of California, Los Angeles, Calif. 90024 2 University of California, Berkeley, Calif. 94720 3 University of California at San Diego, La Jolla, Calif. 92093, U S A
w1. w w w w5. w
Introduction . . . . . . . . . . . . . . . . . . . . . . Linked groups of Pfister forms . . . . . . . . . . . . . . . Treatment of 4-dimensional forms . . . . . . . . . . . . . Pfister neighbors . . . . . . . . . . . . . . . . . . . . . Reduction theorem for linked fields . . . . . . . . . . . . Proof of Main Theorem . . . . . . . . . . . . . . . . .
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w 1. Introduction The notion of an excellent extension of fields was introduced in [ELW1]: a field extension K/F is said to be excellent if, whenever an F-quadratic form q~ is isotropic over K, the K-anisotropic part of q~K(=K| is defined over F. In [ E L W I : (2.2)], we gave a rather long list of known examples of excellent extensions. There is however, one type of very important field extensions which we have not been able to include in this list, namely, extensions K/F where K is a function field F ( { # i } ) of Pfister forms {#i} over F. (We say that K is a Pfister function field over F.) Are such Pfister function fields K always excellent over F? For general base fields F, the answer to this is unknown even in the special Cases (1) {/2i} is a singleton set (cf. [ E L W 1 : Question 6.2]), or (2) all #i are 1-fold Pfister forms. (In the latter case, K is just a multi-quadratic extension of F.) The signifcance about the excellence properties of Pfister function fields is as follows. If all Pfister function fields over a field F are excellent, then F will be an amenable field in the sense of [ E L W I : w4]. If F is an amenable field, we have shown in [ELW1] that many of the major open questions in quadratic form theory can be answered affirmatively for F. It is thus evident that a thorough understanding of excellence properties of Pfister function fields will be a desirable goal. * Supported in part by NSF, and by the Alfred P. Sloan Foundation ** Supported in part by NSF, and by the Miller Institute for Basic Research *** Supported in part by NSF
0020-9910/79/0051/0061/$03.00
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Before we can gather more tangible evidence, it certainly seems premature to conjecture that Pfister function fields are excellent over arbitrary base fields. As a start, it is perhaps prudent to study the excellence question for specific classes of fields F over which the quadratic form theory is better understood; for instance, F a global field, or a field of transcendence degree 1 over the real numbers. For these fields F, we shall show in this paper that all Pfister function field extensions K/F are, indeed, excellent. This will be deduced from the following more general result: (i.1) Main Theorem. Let F be a 1-amenable field with fi(F)=, i=1
(2) F satisfies "Meyer's Theorem", i.e., any t.i. form of dimension > 5 is isotropic over F. (If we assume only (2), but not (1), a slightly modified version of (1.1) holds: see (6.5).) The proof of the Main Theorem occupies the entire paper. Many new techniques are developed to deal with quadratic forms which become isotropic over a Pfister function field. Some of these techniques apply to more general fields, and are thus of interest in their own right. Most notations used in the paper are standard in quadratic form theory. All fields F are assumed to have characteristic 4~2. We write WF for the Witt ring of F, and use the same symbol for a quadratic form and for its image in WF. The fundamental ideal of even-dimensional forms in WF is denoted by 1F, and its n-th power by I"F. The latter is additively generated by P,(F), the set of n-fold Pfister forms ((at, ..., a.)):= + (1,ai)
(ai~P).
i~l
In this paper, we shall use freely the linkage theory of Pfister forms developed in [EL1]. The notion of SAP fields defined in terms of Xv, the Boolean space of orderings on F, will also be assumed throughout (cf. [KRW], [EL2] ). For any field F, u(F) denotes sup {dim q~}, where q~ ranges over F-anisotropic forms with sgn, q~=0 for every e s X v.
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Given an extension field K~_F, r.: WF-~ WK shall always denote the functorial map (sending ~o~--,~pK),and W(K/F) shall denote ker(r.). A K-form is said to be defined over F if there exists an F-form ~0 such that ~ ~ q~K. If we write ~o,,, we shall mean the F-anisotropic part of ~o; if we write (~o~)a,, we shall mean the K-anisotropic part of ~ot(. For convenience of expression, we shall need a slightly modified terminology of "excellence" which unifies the idea of "excellent extensions" in [ELW1], and the idea of "excellent forms" in [K2]. If cg is a class of F-forms, we shall say that K/F is excellent with respect to cg if (q~iOa, is defined over F for all ~p~cs Thus, K/F is an excellent extension in the sense of [ELW1] iff K/F is excellent with respect to all F-forms. Also, an F-form ~0 is an excellent form in the sense of [K2] iff all extensions K/F are excellent with respect to {~o}. For any collection of F-forms {#i}, F({#/}) shall denote the function field of {p~} over F; for properties needed for such function fields, especially Pfister function fields, we refer the reader to [K1], [-K2] and [ELW1]. The terminology of Pfister neighbors follows [K2] (rather than [K~]). For general background on quadratic forms, and other undefined terminology, we refer the reader to [L].
w2. Linked Groups of Pfister Forms The basic result in this section is the "Linked Group Theorem" (2.1), which will play an important role in the subsequent developments in w3 and w4. First let us define what we mean by a linked group (of Pfister forms). Let E be a field. For ~0~P,(E), write ~ for the coset determined by ~0 modulo I"+IE. We say that a subset q~c_P,(E) is a linked group (of n-fold Pfisterforms) if the associated set ~ = {qS: cpecb} is a subgroup in 1"E/1 "+~ E. Note that, if this is the case, then for q~,q~a~(b, there exists ~o3e~ such that q~ +~o/=q~3 (mod I "+ ~E). By [EL~: Th. 4.8], this implies that any two forms in 4~ are linked. Let {q~/} be a subset of P,(E). We say that {~0~} generates a linked group (of Pfister forms) if there exists a linked group 4 ~ P . ( E ) such that {~} generates ~b. If E is a field for which any two n-fold Pfister forms are linked, then, clearly, any subset {~og}_~P,(E) generates a linked group of Pfister forms. In this section, however, E will be an arbitrary field. We can now state the
Linked Group Theorem 2.1. Let {9i} be a subset of P,(E) which generates a linked group of Pfister forms, 4. Let L be the function field E({q~i}). Then q'~_Z WE.q~ i ~_ W(L/E). Moreover, for $6P,(E): $~W(L/E) ,*~ $~q~
for some ~ p ~ .
This lemma may be viewed as a generalization of Kummer theory specialized to multi-quadratic extensions. For, in the case n = l , we have q~i= ( ( - a~))( = (1, - a i ) ) , and the linked group they generate consists o f ( ( - a ) ) where a is a product of the afs. The function field L is just the multi-quadratic extension E({lf~i}). If ~O= ( ( - b)), the conclusion of the Linked Group Theorem (for n = 1) is that b~L2 iff b is a product of ai's (rood/~2). This is, of course, part of the well-known Kummer theory.
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Proof of (2.1). Let ~0e~b. Then q~= ~ol,+ . . . + q~ir(mod In§ for some subscripts i 1, ..., i,. To show that q~e~ WE. q~, we induct on r. If r = 1, we have, in fact, i
q~qo h [ E L l : Cor. to (3.1)]. For r > l , choose a e ~ such that a=~0~2+... + q~ir(mod P + 1E). Then q~- qh, + a (mod P § 1E). This implies that there exists an equation: q~=~oh+a.aeWE, for some ae/~ (see [ELl: Th. 4.8]). By inductive hypothesis, a belongs to ~ WE. q)i. Hence the same is true for q~. i
Now let OEW(L/E). Then ~ becomes hyperbolic already over some E(q)i,, ..., ~%). We may thus assume that the original set {q~} is finite. The rest of the proof will be carried out by induction on t4'1 (the number of isometry classes of q~ecb). If I ~ l = l , then q~={2n-llH}, so L is purely transcendental over E [KI: Prop. 3.8.]. In this case, ~b is already hyperbolic over E [L: p. 255]. Now assume l ~ l > l , say ~0~ is anisotropic over E. Let E~ =E(qg~); clearly I(bg~l--card{~oE : q~eq,} 3, write a = (a, b, a b , . . . ) (after scaling), where all entries are in F. Since F is SAP, we have an F-isometry 2r((a,b))_~2"+l((c)) for a suitable natural number r, and a suitable ceP. After cancellation in W K , we obtain ((a, b)) K - ((1, C))K, SO (a, b, ab)K ~ (c, e, 1)K. Thus a n ~ ( C , c , 1, "")K" This is t.i. over K, so the subform (c, 1, "")K is also t.i. over K. By the inductive hypothesis, (c, 1.... )K~-IH_J_/~0 where /~o is defined over F. Adding on ( c ) K, we get aK~--IH_t_# where /~=g0_I_(C)K is defined over F. QED Remark. If dim a < 3, the lemma is, in fact, true without assuming that F is SAP. (If at = (a, b, ab)K is t.i. over K, then ((a, b))~: is also t.i. over K, and Pfister's LocalGlobal Principle [P: Satz 22] implies that 2 " ( ( a , b ) ) r = O s W K , for some r. The torsion-freeness of i m ( W F ~ W K ) implies that ((a, b)) r is K-hyperbolic, and so a r is K-isotropic). The idea of the proof of (4.1) suggests that one can study the notion of SAP for field extensions instead of just for fields. Most of the known characterizations of SAP fields can be generalized to this "relative" setting. We shall return to this theme in a future work on SAP pairs and SAP triples. In view of the criterion for excellent extensions in [ELWI: (2.1)(e)], we get the following immediate consequence of (4.1).
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Corollary 4.3. Let F~_K be as in (4.1). Then K is an excellent extension ofF. This incidentally leads to a new characterization of a S A P field, in addition to the m a n y other k n o w n characterizations in the literature: Theorem 4.4. For any field F, the following are equivalent: (1) F is SAP. (2) All pythagorean fields K ~_F are excellent over F. (3) All pythagorean SAP fields K ~_F are excellent over F.
Proof ( 1 ) ~ ( 2 ) follows from (4.3), and ( 2 ) ~ ( 3 ) is trivial. (3)~(1). By a t h e o r e m of Craven [Cr], there exists an extension field K _ F such that K is p y t h a g o r e a n and SAP, and the natural m a p X r - , X F is a h o m e o m o r p h i s m . In particular, all orderings on F extend to K, so W(K/F) is torsion (see, e.g., [ELW2: Cor. 3.2]). T o show that F is SAP, it suffices to show (by [EL2: Th. 3.5]) that any pair of 2folds ((a, b)), ((c, d)) (a, b, c, d e f ' ) are stably linked over F. Since K is p y t h a g o r e a n and SAP, (a,b))r, ((c,d)) K are linked over K [EL2: Th. 5.3]. If ((a,b))r~ ((c,d)) K, then 2r (( a, b )) ~- 2" ((c, d)), and we are done. Thus, we m a y assume that (a, b, ab, - c , - d , - c d ) r has K-anisotropic part # of dimension 4. By (3), K/F is excellent, so # is defined over F, say #---(#0)K, where d i m v # o = 4 . F r o m the isometry ((a,b))K_l_ ( -- 1) ((c, d))K-- 21I-I_1_(#0)r, we get
2"((a,b))_l_( - 1) 2r ((c, d)) ~ 2'+ 11H / 2"#o over F (for some r). By [ E L i : Prop. 4.4], 2~((a,h)) and 2~((c,d)) are linked over F. Q E D We shall now set the stage for the criterion for Pfister neighbors over a p y t h a g o r e a n S A P fielld. Recall that, if z is a form over a field F, with 2 " - l < d i m z < 2 ", then z is said to be a Pfister neighor [K2: Def. 7.4] if there exist a~1r peP,(F) such that z l # ~ - a . p , for some F-form #. If this is the case, the form p is uniquely determined (by z), and is called the associated Pfister form of the Pfister neighbor z. The following is an elementary observation on the signatures of a Pfister neighbor. L e m m a 4.5. Let z_l_#~-a.p as above (over any field F ), and let O~u.X F. (1) I f z is indefinite at ~, then I s g n ~ z l < d i m # . (2) If z is not negative definite at ~, then Isgn~(p-z)[ < d i m #.
(Note that dim # = 2" - dim z < dim z.) Proof (1) Since z is indefinite at ~, clearly s g n ~ p = 0 . Thus s g n ~ z = - s g n ~ # and so Isgn~ z[ = ]sgn~ #1 < dim #. (2) First, assume z is indefinite at ~. Then, as above, I s g n , ( p - z ) l = l s g n , zl < d i m # . Next, assume z is definite at c~. By hypothesis, z must be positive definite at ~, and hence so is p. Thus, Isgn~ (p - z)l = 12"- d i m zl = 2" - dim z = d i m #.
QED
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Pfister Neighor Criterion 4.6. Let F~_K where F is S A P and im (WF--* WK) is torsion-free. For any F-form z with 2"-a < d i m z =O for every ~sX~. Assume that (4.8) For all fields E'~_E such that z E, is a Pfister neighbor, the associated Pfister form of r~, is (Pl)E'for some i.
Then there exists a Pi such that Isgn~(pl-z)] d i m z . By [ E L W 2 : Th. 4.18], there exists a field Eo~_E such that i m ( X E o - - * X ~ ) = { ~ l , . . . , ~ } . By Craven's T h e o r e m [Cr], there exists a pythagorean S A P field E'~_E o such that X E, ~ X ~ o is a h o m e o m o r p h i s m . In particular, im(X~, ~ X ~ ) = { ~ , ...,~}. We claim that z~, is a Pfister neighbor over E'. To verify this, we apply the criterion (3) in (4.6). Let flsXe, be such that z~, is indefinite; let fll~ = ~. F r o m [sgn~(p~- z)l > dim z, we see that sgn~, p~ 4:0. Thus, [sgn,, (p~ - z)] = 2" - sgn~, x > dim z,
i.e.,
sgn,, z < 2" - dim ~.
But sgn,, z > 0 by hypothesis, so [sgnr ~e,I = sgn,, z < 2 " - dim ~, as desired.
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We have now shown that z~, is a Pfister Pfister form is (Pi)e, for some i. Choose f l e X not negative definite at fl (since sgn, z>0), ]sgnt~(pi -- Z)E,I = [sgn~, (Pi -- z)[, a contradiction.
neighbor. By (4.8), its associated e, such that fl[v=ei. Surely z E, is so (4.5)(2) implies that d i m z > QED
Remark 4.9. Of course, the proof above shows that we only need the hypothesis (4.8) for those E' which are pythagorean and SAP. Remark 4.10. The proof above made use of Craven's Theorem in [Cr], but when we apply (4.7) for the purposes of proving (6.1), the form ~ in question will be defined over a SAP base field F~_E. In this case, the Pfister neighbor criterion (4.6) can be invoked as long as the field E' above is pythagorean. Therefore, one could have taken E' to be the pythagorean hull of E 0 and avoided the use of Craven's result. In other words, the proof of the Main Theorem can be made completely independent of [Cr]. Under what circumstances can we check the validity of (4.8)? The clue is provided by the Linked Group Theorem 2.1. Combining this theorem with (4.7), we derive the following statement which will be crucial for the proof of the Main Theorem:
Theorem 4.11. Let E be any field, and L=E({q~i}), where {~pi}~_P,(E) generates a linked group of Pfister forms, cb. Let K=L({I~i}), where {#j}_~ U P,,(L). Let z be m>n
an E-form with 2"-1 < d i m z _n are all hyperbolic in R(/~). Thus, the function field K'=R({/t)~ i>n, and all j}) is purely transcendental over R(p) [KI: Prop. 3.8]. Since z is isotropic over K', it must be already isotropic over R(#) [L: p. 255]. But z is positive definite at fl, hence r R is a Pfister neighbor. From this, it is easy to see that dim z > 2"-1 For each ~ X v which extends to L,, we have Isgn~z[2", we can apply (5.1) with E = L , and s = d i m z , and get the desired conclusion for the theorem. For the remainder of the proof, we may thus assume that 2"- 1 < dim z < 2". By (4.11) (applied to E = L , _ 1 and L = L , ) , there exists p~P,(F) in the linked group generated by {/~")}, such that for Z o = Z - p, [sgn~ Zol < d i m z for all ~ X v which extends to L,_ 1. We now apply (5.1) once more, this time to the form z0, with E = L , _ 1 and s = d i m z . The conclusion is that there exists an F-form z l = z o ( m o d ~ WF.#)~ such that Isgn, z 1 [ < d i m z for all ~ X F. By the first i 0 , let E=F(# 1. . . . . #r-x), and R=im(WF--*WE). Let z~W(K/F). T h e n zE~Rc~W(K/E)=Rc~WE.(#,)~ (see, e.g., I-KI: (4.4)]). Since E/F is excellent by (6.1), we have Rc~WE.(I~)e=R.(#r)E, by [-ELWI: (2.11)]. Thus,
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there exists an F-form ? such that zE=(7.#,)~WE. Pulling back to WF, we r--1
have z - 7" #,~ W(E/F), and, by inductive hypothesis, W(E/F) = ~ WF. #i. Heni=l
r
ce re ~ WF.#~.
QED
i=1
In the above argument, we have used the excellence of Pfister function fields K/F to deduce the amenability of F. If, however, we were interested only in the amenability of F, and not in the excellence property of K/F, then a subset of the arguments developed in Sect. 2 and 5 would already suffice. We leave it as an exercise to the reader to extract such a direct proof from the material in Sect. 2 and 5. We shall now give examples of specific classes of fields to which (6.1) and (6.3) apply:
Corollary 6.4. Let F be either (1) a global field, or (2) afield with fi(F)
