higher level reduced witt rings of skew fields

Math. Z. 198, 545-554 (1988) Mathematische Zeitschrift

�9 Springer-Verlag 1988

Higher Level Reduced Witt Rings of Skew Fields

Victoria Powers Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA

Introduction

The purpose of this paper is to extend to skew fields Becker and Rosenberg's theory of higher level reduced forms and reduced Witt rings [4]. We can then show that in the 2-primary case, skew fields give rise to Spaces of Signatures, Mulcahy's abstract setting for this theory [11, 12]. This allows us to extend many results on preordered fields to preordered skew fields, for example we can obtain a "representation theorem" for skew fields.

The notions of a preorder (of level 1) and the reduced Witt ring associated to a preorder were extended to skew fields by Tschimmel [16]. Craven extended the notion of a higher level ordering to skew fields [7], and developed some of the valuation theory which we will need.

The proofs given here follow those of the commutative case, and in some cases no extra work is needed. Occasionally we make use of the commutative case to prove facts "one element at a t ime" by reducing to a commutative subfield of our skew field.

I. Preliminaries

Throughout this paper, D denotes a skew field and b denotes the multiplicative group of nonzero elements of D. We use /~ for the group of complex roots of unity and #,, for the elements of # of order dividing 2m. We define S,(D) and S,(D) as in [7, 2.2], i.e., Sn(D) is the normal subgroup of b generated by 2n th powers and multiplicative commutators, and S,(D) is the subset of D consisting of sums of elements of S.(D), which is a subgroup of b if O(~S.(D) [7, 2.4]. These are the natural generalizations of 2n th powers and sums of 2 r/tla

powers [7, 2.8].

Definition 1.1 (cf, [3, 2.1; 4, 1.1; 16, 1.2]). A surjective homomorphism •: b ~ #~ is a signature of level n if ker ;~ + ker X ~ ker ;~. We write Sgn (D) for all the signatures of D of various levels. A subset T c D is a preorder if T. T c T, ~e+ T c T,

546 V. Powers

and Sm(D)cT for some m. Then, for d6T,, d - l = d - 2 m . d z m - l ~ hence P is a subgroup of b and since P contains the multiplicative commutators and the 2m th powers, b i T is an abelian group of exponent 2 n for some n. We call n the level of T. For a preorder Twe set X r = {x~Sgn(D): Z(P)= 1}.

Remark 1.2. If Z ~ Sgn (D) has level 2" for some n, then ker Z is a normal subgroup o f / ) and D/ker X is cyclic of order 2 n. Thus ker Z is an ordering of level n as defined in I-7, 2.1]. Furthermore, if P c b is an ordering of level 2 n, there is some z~Sgn(D) of level 2 n- 1 with ker Z = P.

Examples of signatures (of all levels) can be obtained using skew fields of twisted Laurent series, see [-3, 3.17] and [7, 2.14].

Definition 1.3. Let T be a preorder. As in [-2, p. 37], M c D is an anisotropic T-module if (a) M + M ~ M, (b) TM ~ M, and (c) M n - M = {0}. If, in addition, (d) M u - M = D , M is a T-semiordering. We say that M is normed if I~M.

Remark 1.4. For any d~b, d M d - l ~ M , since for m~M, dmd-1 = d m d - l m - X m ~ T M c M . Thus M T c M , since for tET and m~M, mt = t t - ~ m t ~ T M ~ M .

Proposition 1.5. Let T be a preorder in D. (i) Each maximal anisotropic T-module is a T-semiordering, and therefore every

anisotropic T-module is contained in a T-semiordering. (ii) A maximal preorder containing T is an ordering of higher level, and in

particular XT 4: ~.

Proof. (i) Let M be a maximal anisotropic T-module. Suppose d~/) and d~ - M . Then M + Td is an anisotropic T-module: properties (a) and (b) are dear, and (c) follows from the fact that P is a group. Thus M + Td = M by the maximality of M, so d~M and hence M w - M = D . Therefore M is a T-semiordering. By Zorn's lemma such an M exists, thus every anisotropic T-module is contained in a T-semiordering.

(ii) Let P be a maximal preorder containing T. Suppose d~D is such that d2E P but d ~ P. If - d ~ P then as above there would be a larger preorder containing P, contradicting the maximality of P. Therefore, if d2Ep, then d ~ P u - P . Thus, as in the proof of [-7, 2.63, we can show that /)//5 is cyclic of order 2 k for some k. Hence P is an ordering of higher level. By Zorn's lemma such a P exists, therefore Xr =~ g~.

Proposition 1.6. I f T is a preorder of level 2 k for some k, then T= fqx~xT ker Z.

Proof. An easy generalization of [-7, 2.8].

Remark 1.7. Unfortunately, the proof of 1.6 works only in the 2-primary case. Since this is one of the key results needed, our main theorems will only hold in the 2-primary case.

Definition 1.8. As in [4, 1.5], an r-dimensional T-form p is an r-tuple (al T, ..., arT), ai~F~f)/T. For convenience we Write p = ( a 1, ..., at) and dimp = r. The empty form p = ( ) has dimension 0.

The set of elements Dr(p) represented by p = ( a l . . . . , a~) is {~,ait~: t~T} .

Higher Level Reduced Witt Rings of Skew Fields 547

A T-form p = ( a l , ..., at) is T-isotropic if there exist ti, ..., treT, not all 0, such that ~ at t, = 0, and T-anisotropic otherwise.

Lemma 1.9. Suppose T is a preorder of level n and S c D is such that S + S c S, TS c S, and S c~ -S~e {0}. Then S = D.

Proof As in [7, 2.8].

Theorem 1.10 (cf. [4, 1.6]). The following are equivalence for a T-form p = (a i . . . . . a~):

(i) p is T-isotropic. (ii) DT(p) c~ -- DT(p) * {0}.

(iii) Dr(p)=D. (iv) DT(p) is not contained in any T-semiordering.

Proof The proof of (i) iff (ii) is the same as the commutative case, and (iii) implies (ii) and (ii) implies (iv) are clear. Suppose (ii) does not hold, then DT(p)n--DT(p)= {0}, SO DT(p) is clearly an anisotropic T-module. Hence, by 1.7, DT(p) is contained in a T-semiordering. Thus (iv) implies (ii). If ~a~ti=O with t l ~ 0 , w e have alEDT(p)C~--DT(p). Then, by 1.9, DT(p)=D , proving (i) implies (iii).

II. Valuation Theory

A subring A c D is a valuation ring if, for any deD, d or d - l e A . If, in addition, d A d - l = A for all deD then we say A is invariant. Unless otherwise stated, all valuations rings are assumed to be invariant. The basic definitions and results for invariant valuation rings in skew fields can be found in [15]. Results for noninvariant valuation rings can be found in [10]. We use v for the valuation, F for the value group (in the case of noninvariant valuations), A for the groups of units, I for the maximal ideal, and d for the residue class skew field.

Following the notation of [3], we say a homomorphism X: b ~ # is compatible with a valuation A, written A ,-~ Z, if 1 + I c ker Z- Then, as usual, the equation )~(ti)=Z(u ) yields a well-defined homomorphism )~: d ~ / l . A preorder T is fully compatible with A, written A,-~sT, if 1 + I c T. We say T is compatible with A, A ~ T, if A "~X for some )~eXr. f2(T) denotes the set of valuations compatible with T. Set T ~ = T ( I + I ) , then it is clear that T ~ is a preorder if 0r ~. If T ~ is a preorder, then v ~$ T ~.

Proposition 2.1 (cf. [3, 2.5]). Suppose v is a valuation and Z: b--* p is a homomorphism such that v',~)b Then zeSgn(D)/ff )~eSgn(d).

Proof Same as the commutative case.

Proposition 2.2. Let T be a preorder of D. (i) I f vef2(T), then T ~ is a preorder.

(ii) (cf. I-4, 2.3(ii), 2.5]). A~f2(T) iff T,, the image of Tc~A in d, is a preorder, called the pushdown of T along A.

Proof (i) We need only show that 0 r ~r~ + ~ . Suppose t 1 (1 + Xl) + t2 (1 + x2) = 0, where t~e T and x~eI. Pick x e X T with v,,~ X, then applying )~ to the above equation yields ker Z + ker)~ q~ ker X, a contradiction.

548 V. Powers

(ii) is proven exactly as in the commutative case.

Lemma 2.3. Suppose S is a T-semiordering and A is a (possibly noninvariant) valuation ring with 1 + S~ S. Then ~F is a preorder in d and T ~ is a preorder in D.

Proof It is clear that Tis a subgroup of d, and since S , ( D ) c T, 2 , (d)cN, (O)c T. Suppose t l, tEe T, then t l, t2 e T n A and t~ + t2~A. If t t + t 2 E l , then t l = - ta(1 +x) for some xeI. But then _ t z ( l + x ) ~ F c S and since 1+ITS , tz(l+x)ES, a contradiction. Thus T+ T c T and so T is a preorder. The proof that T" is a preorder is similar to the proof of 2.2(ii).

Next we look at valuation rings associated with signatures and semiorderings of D. For a subset H o D let A(H) be the convex hull of Q w.r.t. H, i.e., A(H)={d~D: r+d~H for some rEQ +} and let I(H)={d~D: r-t-d~H for all reQ+}. For zeSgn(D), we use A(Z) to denote A(kerz), and I(Z) to denote I (ker Z)-

Lemma 2.4. (i) Suppose Q + c H and H is closed under addition and multiplication. Then A(H) is a subring of D and I(H) is a 2-sided ideal of A(H). In particular, for a preorder T, A(T) is a subring of D with a 2-sided ideal I(T).

(ii) (cf. [7, 3.5]). Suppose H o D and d H d - l c H for all deD. Then for all d ~ D, d A (H) d- 1 = A (H) and d I (H) d - 1 = I (H).

Proof (i) This is straightforward, see [7, p. 82]. (ii) For aEA(H) and r~Q + with r++_aeH, r+dad- l=d(r+_a)d- lEdHd -1

c H. Thus d a d - l e A ( H ) and so d A(H)d -1 cA(H) . A similar proof applies to 1(I-I),

Proposition 2.5. (i) For zESgn(D), A(Z) is a valuation subring of D with maximal ideal I(Z), A(Z)~)~, and ~, the pushdown of Z along AO0, is the signature of an archimedean order.

Proof By 2.4, A00 is a subring of D with ideal I(Z). As in [7, 3.6], if a(~A()O consider K = Q ( a ) c D and A~c=A(kerznK)=AO~)nK. Then aq~AK, hence by the commutative case a- leA~cA()O. Thus A(Z ) is a valuation ring, and a similar proof shows that I00 is the maximal ideal. Using the same type of argument, we can show that if aeA() 0 then aa~ker Z, hence )~ is an archimedean ordering of A ()Off 00.

We now want to show that for a T-semiordering S, A(S) is a valuation ring. As in the commutative case, we cannot show directly that A(S) is even a ring, but we show that it is a localization of the ring A(T). Our proof makes use of the result for the commutative case [2, 2.1].

We fix a normed T-semiordering S and for convenience set A = A (S), I = I (S), B=A(T), and J = B n I . We start by using the commutative case to obtain a description of the elements of A (S).

Proposition 2.7. For any dEA there is some a~B and q~(B\J)c~ T such that d =aq -1.

Proof Let F be the center of D and let K=F(d), a commutative subfield of D. Set Tr= T n K and SK=SnK, then A(SK)=Ar~K, A(TK)=BnK, and I(SK)


=Ic~K. From the proof of [2, 1.2], we have d=aq -1 for some a~A(TK)cB and q~(A(TK)\(A(TK)c~I(Sr))n Tr. Then q~TK=T, q~A(TK)=B, and qr hence q•J. Thus q~(B\J)c~ T.

Theorem 2.8. A (S) is a valuation ring with maximal ideal I (S).

Proof By 2.5, B is an invariant subring of D and J is an invariant B-module. If A is a ring, then as in 2.5 we can use the commutative case to show that A is a valuation ring with maximal ideal I.

Let Q=(B\J)c~T, then Q is multiplieatively closed: For a, b~Q, we can find r, s e Q + such that r - a and s - b are not in S, and therefore a - r and b - s are in S. By definition, a, beT, hence a+r, b + s e T and we have a b - r s =�89 Thus rs-abq~S and so ab(~J, hence a b ~ Q.

By 2.5, Q is invariant, hence by [10, p. 39] BQ-1 is a ring with right ideal jQ-1. By 2.7, A c B Q -1, thus we need only show that B Q - l c A . Suppose

1 qeQ, then q~T and r-q(~S for some r e Q +, hence q+r~S. Then - + q - l ~ S ,

r thus q-leA(S). Hence Q-1 cA(S) and so BQ -1 cA(S). Therefore A(S)=BQ -1 is a ring, and we are done.

We end this section by looking at the relationship between forms over a preordered skew field with a compatible valuation and forms over the (preordered) residue class skew field. We do this from the point of view of Spaces of Signatures (SOS). We will now need to make use of 1.8, hence for the rest of this paper we assume all preorders have level 2 k for s o m e k.

Recall that a SOS is a pair (X, G) where G is an abelian group of finite even exponent and X is a subset of Horn(G, #) which satisfies certain axioms (for details see [11]). A form over (X, G) is an n-tuple f = ( a l . . . . . at) where ai~ G. If a~X then ~ a(aO is denoted by a(f). Two forms f and g are equivalent, f - g , if a(f)=a(g) for all a~X. If, in addition, d i m f = d i m g , then they are isometric, f ~ g. For a nonempty form f the set D ( f ) = {a ~ G: f ~ (a, c2, ..., er) for some cisG} is called the represented set of f .

Proposit ion 2.9. Let TeD be a preorder. Then in the language of [11, 1.1], (XT, b / T ) is a pre-SOS.

Proof We must show that Axioms So through S a hold. The proof of So is the same as in the commutative case [4, 17(i)], $2 is obvious, and S 3 is 1.8. The proof that XT is a closed (in the compact-open topology) subset of D/T is the same as the proof for the field case [4, 1.4(i)] and thus $1 holds.

Remark 2.10. In order to show that (XT, D/T) is a SOS we need to show that the abstract and concrete notions of representability are the same, i.e., that for a T-form p, D(p)=DT(p). This will take most of the work of the rest of this paper.

Definition 2.11 (cf. [11, 2.6]). A pre-SOS (X, G) is a group extension of a pre-SOS (X', G') if there is a group injection G' ~ G such that

X = {a~Hom(G, #): al~.~X'}

(where we identify G' with its image in G).

550 v. Powers

Proposition 2.12 (cf. [4, 2.6], 13, 2.3]). Suppose T is a preorder and veX(T) . Then (XT~, {)/T~) is a group extension of (XT-, d/7~).

Proof Using 2.2, the proof for the commutative case holds.

Definition 2.13. Let veQ(T). Using 2.11 and following [11, 2.6(ii)], we fix a set of coset representatives {c~} of F modv(T) and define residue class forms: for a form p over XTv, and each 7eF/v(T), we have a form p~ over X~r such

t h a t p = (~c~p~. The p~'s are uniquely determined up to permutation of their entries, only finitely many are nonempty, and dim(p)= S~ dim(p~). In the notation of [4, 2.10], if ae / ) with v(a) v(7") = c~, we set pa=C~p~ and fia=pr

HI. The Isotropy Principle

The goal of this section is to generalize the isotropy principle for preordered fields [4, 3.3]. This will allow us to show that the abstract and concrete notions of representation are equivalent.

Definition 3.1 (of. [2, p. 42]). Suppose S is a normed T-semiordering and A is a (possibly noninvariant) valuation ring in D. We say A and S are compatible, A ,-~ S, if (1 + I) S c S and A and S are weakly compatible, A ~wS, if (1 + I)(A n S) c S. We write S for the image of S n A in d.

Proposition 3.2 (cf. [2, 1.2, 2]). The following are equivalent for a nontrivial (possibly noninvariant) valuation ring A c D and a normed T-semiordering S: (i) A ~-~S, (ii) S is a 7"-semiordering, (iii) 1 + I c S, (iv) A(S) c A.

Proof Since l e A n S , (i) implies (iii) is clear. Suppose l + I c S , then by 2.4, is a preorder, and it is easy to check that S + S c S, and S u S = D. If de E ~ - S ,

where 4=0, then by 1.10, S=d, which implies - l e S ~ a contradiction. Hence S n - S = {0} and thus Sis a T-semiordering. This proves (iii) implies (ii). Suppose

is a T-semiordering and ( l + x ) u r for some x e I and ueAc~S. Then ( 1 + x) u e - S, hence - ~ e S, but t7 e S also, a contradiction. Thus (1 + I) (A c~ S) c S, which proves (iii) implies (i). The proof of (iii) iff (iv) is the same as the commutative case [2, 2.2].

A valuation ring in a skew field induces a topology on the skew field, see [10, w Suppose A is a rank 1 valuation in D, then A induces an absolute value on D, and thus /), the completion of D w.r.t, the topology induced by A, is again a skew field [5, Prop. 6, w 3, No. 2].

Lemma 3.3. Let A be a rank 1 valuation in D, and let ID be the completion of D with respect to A. Given meN, then for any xeI , there is some yeD such that 1 + x = ym.

Proof Fix x e l and let F be the center of D, K=F(x) , and A r = A n K . Let FK denote the value group of AK, etc. Since A has rank 1, F is archimedean [15, Lemma 4, p. 4] and hence FK is also. Thus AK has rank 1. Let / ( be the completion of K with respect to AK, then / ( satisfies Hensel's lemma, hence 1 + x = y " for some ye /~c /5 .


As in the commutative case, a T-semiordering S imposes a total ordering on (D, +): a<sb iff b - a t S . This induces a topology on (D, +). We also have the topology induced by A(S) which is a skew field topology by [10, 6.3]. The proof of [2, 1.5] generalizes to show that these topologies are the same, in particular, the topology induced by < s is a skew field topology.

Proposition 3.4 (cf. [2, 2.5]). Suppose A is a rank 1 valuation and A~wS. Then A~S .

Proof Since A ( S ) c A + D , A(S) and A induce the same topology on D [10, 6.2], which, by the above remark, is the topology induced by the total ordering <s- Then, as in [2, 1.6], we can show that ~, the closure of S in /3, is a 7"- semiordering, where T denotes the completion of T in /3. Suppose T has level n, then by 3.3 we have (1 + I) S c D n Z,(/3) S c D c~ S = S, hence A ~ S.

We would now like to show that the pushdown of a semiordering S along A(S) is actually an ordering. To do this, we need to generalize some of Prestel's results on quadratic semiorderings [14, w We fix a Iz(D)-semiordering S and let < be the total ordering on D induced by S. Note that if 0 < a then 0 < d Z a and 0 < a d 2 for any diD.

Lemma 3.5 (cf. [14, 1.18]). (i) IfO<a, then 0 < a -1 (ii) I f O<a<b, then aba<ab 2.

(iii) I fO<a<b and atS2(O), then a 2 < b 2.

Proof (i) I f 0 < a , then O<aa-2=a -1 (ii) If 0 < a < b, then by (i), 0 < a - ~ and 0 < ( b - a)- ~. Then we have 0 <

( (b -a ) -1 +a-1)(b-a) -1 = a -1 b2-a -1 ba. Thus O < a b 2 - a b a t S , hence aba <ab 2.

(iii) Since a- l t I2 (D) , by (ii), al(aba)<a-l(ab2), or ba<b 2. Since a<b and at I2(D ), a2 <ba. Then aZ <ba<b 2.

Proposition 3.6 (cf. [-14, 1.20]). I f S is an archimedean 12(D)-semiordering then S is an ordering.

Proof Clearly we need only show that S. S c S. In fact, it is enough to show that a b t S or b a t S for a, btS: If abtS , then by 3.5(i), ( a b ) - l = b - l a - l t S , hence b a = b2(b-la - ~)aZtS. As in the proof of the commutative case, using 3.5(iii) we see that ab+ba tS .

Theorem 3.7. Suppose T is a pushdown of S along A = A (S),

Proof By 3.2, ( l + I) (A n S) c then use the same argument a 2 (fi n S) c A ~ S for every

If abq~S, then - a b t S , hence batS .

preorder and S is a T-semiordering. Then S, the is a ordering of level 1 in d.

S and / i n S is additively closed by 2.1. We can as the commutative case ([2, 1.2]) to show that atA. Hence I2 (d ) . ScS and thus S is a

I2(d)-semiordering. It follows from the definition of A(S) that S is archimedean, therefore S is an ordering of level 1 by 3.6.

Lemma 3.8. Let p = ( 1 , al, ..., at) be an anisotropic T-form. Then there is a T- semiordering S containing Dr(p) such that A(S)=A(z) for some z tXrv , where v is the valuation of A (S).

552 v. Powers

Proof There is a T-semiordering S containing DT(p) by 1.10, in particular S is normed. By 3.7, S, the pushdown of S along A(S), is an ordering. Let a be the signature of S, then by 2.9 there is some Z~XTv, such that )~=a. Then, as in [3, 3.5] it is easily shown that A(Z)= A(S).

Theorem 3.9 (cf. [4, 3.1]). Suppose aieS, i= 1, ..., r and vs(al)r ) for some i. Then there is a valuation ring B # D with valuation v such that

(i) A (S) c B (ii) (1 + I) at c S for all i

(iii) v(ai)6v(T) for some i.

Proof Let F = {B~} be the family of overrings of A(S) which satisfy the following property: there is an i, l<=i<r, such that a4=tu for any t ~ T and ue/)~. Note that each B~ is a (possibly noninvariant) valuation ring. By [10, 1.3], the B~'s are linearly ordered by inclusion so that B = U B~ is a ring, which clearly satisfies (i) and (iii).

Given B ~ F and d~b, we claim dB~d- l~F . Suppose not, then for each i=1, . . . , r there is some wz, a unit in dB~d -s, and t~ET such that a~=hw t. Clearly the group of units of dB, d- ~ is precisely d[3~d- 1. Hence a~ = t t du~ d- s for some u ie[3~, but then we have a t = t t d u t d - ~=siui where st =h(dutd -s u~a)eT, which implies B~r a contradiction. Thus d B~d-aeF. Since the conjugates of the B~'s are again B~'s, B is an invariant valuation ring.

By [16, Thm. C, p. 13], a valuation ring in a skew field with no proper overrings is a rank 1 valuation ring. Using this fact, along with 3.2 and 3.4, the remainder of the proof is exactly as in the commutative case.

Definition 3.10 (cf. [4, 3.2]). For al . . . . , a,~D, let X(T, al, ..., at)= {Z6XT: vx(at) =vx(aj)modvx(~F) for all i and j}, where v x denotes the valuation of A(Z ). Y2(T, aa . . . . . a,)={v~O(T): v(al) ~-v(aj) for some i +j}.

Theorem 3.11 (cf. [4, 3.3]). A T-form p = <as . . . . . a,> is T-isotropic iff (i) p is )~-indefinite for all ZeX(T, aa . . . . . a~), and

(ii) p is T~-isotropic for all v~O(T, al, ..., a,).

Proof Using 3.8 and 3.9, the proof of the theorem is as in the commutative case.

IV. Higher Level Reduced Witt Rings

Definition 4.1 (cf. [7, 4.8; 4, 4.1]). Suppose T is a preorder in D. As in [12, 1.43, each a~/} induces a continuous function ~ from XT to the discrete space C via c~(a)=a(a) for each a ~ X r . Let C(X T, C) denote the ring of continuous C-valued functions on XT. The subring WT(D ) of C(XT, C) generated by the ors is called the reduced Witt ring of T. For a T-form p = <al . . . . . at>, we write

r

/~ = ~ ai. We say two T-forms p and z are isometric, written p ~z, if/~ = ~ in 1

Wr(D) and dim p = dim z.


Proposition 4.2 (cf. [4, 4.493). Suppose p and z are T-forms, ~=~ on Wr(D), and dim p < dim ~. Then z is T-isotropic.

Proof By [11, 1.43, d i m p = d i m z ( m o d 2 ) and by [11, 2.7], for any a, pa=% on X Tv, for any v e f2 (T). Using these facts plus 3.11, the proof for the commutative case generalizes.

As in [-4, 4.10], an immediate consequence of 4.2 is

Corollary 4.3. I f p ~ z, then DT(p)= DT(Z ).

Proposition 4.4 (cf. [4, 4.11]). Suppose b and l + b are in D, then (1, b ) ~ ( 1 +b, (b2 +b)(l +b2)- l ) .

Proof Same as the commutative case.

Corollary 4.5. Suppose a, b and a + b are all in D, then ( a, b ) ~ ( a + b, (b a- X b +b)(1 + a -1 b2)- 1).

Proof Substitute a - 1 b for b in 4.4. We can now show that the abstract and concrete notions of representation

are the same:

Proposition 4.6. XEDT(p) iff xED(p).

Proof If XEDT(p), then x = ~ a i t i for some tieT. Then repeated applications of 4.5 show that ( a i t i , . . . , a r t r )~ ( x , ca, . . . ,cr) for some ci~T Thus x6D((aiti , ..., art~))=D(p). If x6D(p), then X~DT(p) by 4.3.

As in [4, 4.12], Witt's chain-equivalence theorem for preordered skew fields now follows:

Theorem 4.7. For T-forms p and z with d i m p = d i m z , p ~ z iff z can be obtained from p by finitely many changes of the form

(i) ( a ) ~ ( t a ) , teT, (ii) (a, b) ---, (b, a )

(iii) (a, b) --, (a+b, (b a -1 b+b)(1 + a -1 b2) - 1).

Theorem 4.8. For a preorder T, (XT, D/]F) is a SOS.

Proof. By 2.10 we must show that $4 holds. By 4.6, we need only show that S 4 holds substituting Dw(p) for D(p) and this is immediate.

Now we can apply all of the results of [11, 12] to (XT, D/]F) for T a preorder of a skew field D. We conclude by stating the Representation Theorem for skew fields.

Definition 4.9 (cf. (i) (cf. [12, 4.1]). A preorder T is a quasifan if (XT, D/~') is a group extension of a SOS (X, G) where G is a cyclic group or a direct sum of 2 cyclic groups.

(ii) (cf. [4, 6.1; 12, 4.7]). Suppose T c T,, then FeC(XT, C) is represented over XT, if F restricted to X~- is ~3 for some T-form p.

Theorem 4.10.. F~C(XT, C) iff F is represented over X~ for all quasifans 7"~ T such that ID/TI < o0.

Proof This is [12, 5.3].

554 V. Powers

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Mfinster, 1981

Received October 3, 1986

higher level reduced witt rings of skew fields

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