the Uniqueness of

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On the Uniqueness of Real Closures and the Existence of Real Places

M A N F R E D K N E B U S C H ( S a a r b r ü c k e n )

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a is the restriction o f r to K, i f a = xocp^. I f K a n d L are fields, this means that cp is c o m p a t i b l e w i t h the orderings c o r r e s p o n d i n g to a a n d T. I f p is a p r i m e i d e a l o f ^ ( A T ) l y i n g m i n i m a l l y over A{cp) then there exists a ( m i n i m a l ) p r i m e i d e a l q o f W(L) w i t h

^>^1(q) = p (cf. [5] C h a p . I I . § 2 n o 7 P r o p . 16). T h i s r e m a r k i m m e d i a t e l y i m p l i e s

P R O P O S I T I O N 2A. A signature a of K can be extended to a signature of L with respect to cp if and only if <j(A((p)) = 0.

§3. The Transfer Map

N o w assume i n a d d i t i o n that cp:K-+L is finite a n d separable. I n o u r case, this means that i n the d e c o m p o s i t i o n L = Ll x ••• x Lr o f L as a A^-module c o r r e s p o n d i n g to the d e c o m p o s i t i o n K=K^l x ••• x K„ any L^t is either zero o r a p r o d u c t o f finite separable field extensions o f K^t. F o r later convenience I r e c a l l the d e f i n i t i o n o f the regular trace Tr = Tr(p f r o m L to K ([3], p . 397, [7]). W e identify the r i n g E n dx( L ) o f AT-linear e n d o m o r p h i s m s o f L w i t h Uom^K(L, K)®^KL. T h i s is possible since L is a finitely generated projective AT-module. F o r a n y x i n L we denote b y L(x) the ^ - l i n e a r e n d o m o r p h i s m j>h->jty o f L. W e further denote b y e: E n d * ( L ) -> K the e v a l u a t i o n m a p f®x\->f(x) {fin H o m ( L , K), x i n L } . T h e regular trace is defined b y

Tr(x) = e(L(x)).

A s is w e l l k n o w n ( l o c . cit.) the A^-bilinear f o r m Tr{xy) o n L is n o n singular. F r o m this one sees i m m e d i a t e l y ( c f [6]) that for a n y space (E9 B) over L the Af-module E e q u i p p e d w i t h the b i l i n e a r f o r m TroB: Ex E-> AT is a space over K. W e thus get a n a d d i t i v e m a p ([18])

Tr*:W(L)-+W(K).

s e n d i n g { ( £ , B)} to { ( £ , TroB)}. T h i s " t r a n s f e r - m a p " is related to (p+\W(K)^> W(L) b y the f o l l o w i n g " F r o b e n i u s l a w "

P R O P O S I T I O N 3.1. ([18]).

x-Tr* (y) = Tr*((p* (x) y)for x in W(K) and y in W{L).

I n fact, f o r spaces (E^u B^x) over K a n d (E², B²) over L the c a n o n i c a l m a p f r o m El®KE1 to (L®KEi)®LE2 is a n i s o m o r p h i s m o f spaces w i t h respect to the f o r m s B^l®^K{TroB¹) a n d Tro(B[®^LB²) where B[ is o b t a i n e d f r o m B^t b y base e x t e n s i o n .

W e denote the image Tr*(W(L)) b y M{cp). T h e F r o b e n i u s l a w i m p l i e s

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C O R O L L A R Y 3.2. M(q>) is an ideal of W{K) and A {cp) M((p) = 0.

T h e transfer m a p is c o m p a t i b l e w i t h base extensions. T h i s is the content o f the f o l l o w i n g " M a c k e y l a w " , whose i m p o r t a n c e has been revealed b y D r e s s i n [6].

P R O P O S I T I O N 3.3. Assume cp\K-*L and ^:K-+ M are homomorphisms between (semisimple) rings and that cp is finite and separable. Then the diagram

is commutative.

Indeed, b y erasing the letters " W" a n d the asterics, one o b t a i n s a c o m m u t a t i v e d i a g r a m , as f o l l o w s easily f r o m the d e s c r i p t i o n o f the regular trace a b o v e . S t a r t i n g f r o m this fact the p r o o f o f P r o p o s i t i o n 3.3 is s t r a i g h t f o r w a r d .

W e shall often write Tr^L/K instead o f Tr⁹⁹ w h e n there is n o d o u b t as t o w h i c h m a p p i n g cp-.K-*L is b e i n g c o n s i d e r e d .

Remark. C l e a r l y the statements o f this section r e m a i n true f o r the W i t t rings o f a r b i t r a r y c o m m u t a t i v e rings i n the sense o f [12], i f cp\K-+L is finite etale, i n o t h e r w o r d s , i f L is a finitely generated projective ^ - m o d u l e a n d a projective L ®^KL - m o d u l e ([8] P r o p . 18.3.1, p . 114).

§4. Proof of the Uniqueness Theorem 1.3

T h r o u g h o u t this section L/K w i l l be a finite e x t e n s i o n o f fields o f characteristic zero. D e n o t e b y 7>*(1) the value o f the u n i t element (1) o f W(L) u n d e r the transfer m a p Tr*: W{L)-+ W(K) w i t h respect to the i n c l u s i o n i:K-*L.

L E M M A 4 . 1 . For any signature a of K we have a(Tr* ( 1 ) ) ^ 0 . The signature a can be extended to L if and only if o(Tr*(\))>0.

Proof W e p r o c e e d b y i n d u c t i o n o n n = \_L:K']. F o r n=l the assertion is t r i v i a l . A s s u m e n > 1. I f a c a n n o t be extended to L t h e n b y P r o p . 2.1 we have a (A (i)) # 0 a n d hence b y C o r . 3.2 a ( A / ( / ) ) = 0, i n p a r t i c u l a r <j(Tr* (1)) = 0. N o w we assume that c has at least one e x t e n s i o n T i n S i g n L. T h e n

W(L) T>*4

W(M®^KL)

W(M) W(K)

°(Tr*(l)) = r(UTr*(l)).

B y the M a c k e y l a w 3.3 the d i a g r a m

W(L)

W(K)

W(L(g)^KL)

J T I - ' L ® ! . / ! .

W{L) I*

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is c o m m u t a t i v e . H e r e L®KL is c o n s i d e r e d as a n extension o f L by \®i\L = L®KK-+

-+L®^KL. T h i s extension is L - i s o m o r p h i c w i t h a p r o d u c t £^L x E² x ••• x E^t where E^t =L a n d E ² , E ^t are algebraic field extensions o f L w i t h degrees s m a l l e r than n.

T h u s

* ( 7 > 2^{/ x}( l ) ) = I t ( T r *^{/ L}( l ) ) . i = 1

T h e first s u m m a n d equals 1 a n d the others are ^ 0 by the i n d u c t i o n hypothesis, w h i c h shows 0 " ( 7 > *^{/ K}( l ) ) > O . q.e.d.

F o r later convenience we r e c a l l the f o l l o w i n g

E X A M P L E 4.2. A s s u m e L = K(yja) w i t h some a # 0 i n K. T h e n a signature <x o f AT is extendable to L i f a n d o n l y i f a(a)=\. Indeed, w i t h o u t loss o f generality assume L^K. 7>*(1) is represented b y the space L over K w i t h b i l i n e a r f o r m Tr(xy). T h i s space is i s o m o r p h i c to (2)1 (2a) ( C o n s i d e r the basis 1, yja).

F o r the r e m a i n d e r o f this section R denotes a real closed field a n d g denotes the signature o f R.

L E M M A 4.3 Assume cp is a homomorphism from K into R such that

e( < P * 7 > * ( l ) ) > 0 .

Then cp can be extended to homomorphism from L into R.

Proof L e t a denote the signature g°cp* o f K. T h e tensor p r o d u c t R®^KL construc- ted f r o m cp\K-+R a n d the i n c l u s i o n i:K-*L has a d e c o m p o s i t i o n

R ®KL = Ei x ••• x Et

i n t o fields. W e regard R®KL as a n extension o f R by the m a p l ® / f r o m R = R®KKto R®^KL. F r o m the M a c k e y l a w 3.3 we o b t a i n as i n the p r o o f o f l e m m a 4 . 1 :

°(TrlK(l)) = QoTr*9L/R(l)= £ QoTrtj/R(l).

i = i

N o w a c c o r d i n g to (1.1) a n y E^} is / ^ - i s o m o r p h i c either to R o r to the algebraic closure R o f R. Since W(R)^Z/2Z is a t o r s i o n g r o u p , Tr*^m(\) = 0. T h u s we see that cr(Tr*^/K(\)) equals the n u m b e r o f c o m p o n e n t s Eⁱ w h i c h are i s o m o r p h i c to R. R e c a l l that Ex,..., Et f o r m a full system o f inequivalent field composites o f L a n d R over K ([11] C h a p . I, § 1 6 ) . T h u s it f o l l o w s that a ( 7 r *^{/ J C}( l ) ) is the n u m b e r o f different h o m o - m o r p h i s m s f r o m L to R w h i c h extend cp. Since b y a s s u m p t i o n cr(Tr*^/K(\))>0, the p r o o f is c o m p l e t e .

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P R O P O S I T I O N 4.4. Assume L is ordered and cp:K->R is an order preserving homomorphism. Then there exists an order preserving homomorphism i/r. L - » R which extends cp.

Proof, i) L e t a denote the signature o f K c o r r e s p o n d i n g to the r e s t r i c t i o n o f the o r d e r i n g o f L to K. T h e a s s u m p t i o n that cp is order preserving means (J = Q°(p^. Since o c a n be extended to L we k n o w f r o m L e m m a 4.1 that c r ( 7 > * ( l ) ) > o a n d thus by L e m - m a 3.4 that there exists at least one h o m o m o r p h i s m f r o m L to R w h i c h extends cp.

ii) L e t \jj^r denote the different h o m o m o r p h i s m s f r o m L to R w h i c h extend cp. W e must s h o w that at least one o f the ^ is o r d e r preserving. A s s u m e the c o n t r a r y is true. T h e n for any 1 ^ / ^ r , we have a n element at>0 i n L such that i / ^ ( at) < 0 . B y example 4.2 the o r d e r i n g o f L can be extended to the field M—L{Ja^x,yja^r).

B y p a r t i) o f o u r p r o o f there exists a h o m o m o r p h i s m / : M - » 7 ? w h i c h extends cp.

C e r t a i n l y x(a) = x(\/ai)2 > 0 f °r T h u s x | L can not c o i n c i d e w i t h any o f the

ij/i9 w h i c h yields the desired c o n t r a d i c t i o n . q.d.e.

A f t e r these preparations the p r o o f o f T h e o r e m 1.3 is easy. F r o m P r o p o s i t i o n 4.4 a n d an a p p l i c a t i o n o f Z o r n ' s l e m m a it f o l l o w s that the given order preserving h o m o - m o r p h i s m cp\K-> R can be extended to a h o m o m o r p h i s m \j/ f r o m the real closure Fto R. W e still must s h o w that \j/ is the o n l y extension o f cp f r o m F to R. F o r any x' F-* R extending cp the images / ( F ) a n d ij/(F) b o t h c o i n c i d e w i t h the algebraic closure o f A ^ i n R. T h u s there exists a n a u t o m o r p h i s m X o f FjKW\\hx = ^°^- O f course A is order pre- serving. A s s u m e k is not the identity. T h e n we can find some x i n F w i t h x<X(x).

A p p l y i n g X repeatedly to this i n e q u a l i t y we o b t a i n

x < X(x) < X2(x) <••• < Xn(x)

for a l l n, w h i c h is i m p o s s i b l e since Xm(x) = x for some m (depending o n x). T h i s c o m - pletes the p r o o f o f T h e o r e m 1.3.

§ 5. A Trace Formula for Signatures

I w a n t to t h r o w some m o r e light o n L e m m a 4.1. W e first return to P r o p o s i t i o n 4.4.

W e denote b y F a fixed real closure o f L w i t h respect to the given o r d e r i n g . B y T h e o - r e m 1.3 any order preserving extension ij/:L->R o f cp c a n be further extended to F.

T h u s the uniqueness statement i n T h e o r e m 1.3, a p p l i e d to F/K, shows that there is a unique order preserving \jj\L-+ R w h i c h extends cp. W e o b t a i n the w e l l k n o w n

C O R O L L A R Y 5.1. L e t L/K be a finite algebraic field extension a n d cp:K-* R a h o m o m o r p h i s m f r o m K i n t o a real closed field R. D e n o t e b y Q the signature o f R a n d by a the i n d u c e d signature Q^cp^ o f K. T h e n the signatures T o f L extending a corres- p o n d bijectively w i t h the h o m o m o r p h i s m s if/.L^R extending cp v i a T = QO[j/^.

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W e stay i n the s i t u a t i o n o f this c o r o l l a r y . D e n o t e b y S the set o f a l l ij/.L^R ex- t e n d i n g cp. R e p e a t i n g the p r o o f o f L e m m a 4.3 w i t h a n a r b i t r a r y element £ o f W(L) instead o f the unit element we o b t a i n

w i t h the c o n v e n t i o n that i f S is e m p t y this s u m is zero. C o r o l l a r y 5.1 n o w i m p l i e s

P R O P O S I T I O N 5.2 Let L/K be a finite algebraic field extension and a a signature ofK. For any c in W{L)

^ ( T r

^L

V ( 0 ) = Z r ( 0 ,

where x runs through all signatures of L extending a.

T h i s p r o p o s i t i o n generalizes L e m m a 4.1.

§6. Existence of Real Places on Function Fields

In this section R denotes a real c l o s e d field, K a real finitely generated field exten- s i o n o f R o f transcendency degree 1, a n d S a fixed real c l o s e d field extension o f R (e. g. S=R). W e are interested i n places cp:K-+ Suco over R, i . e. w i t h cp the i d e n t i t y o n R. T h e a i m o f this section is to p r o v e the f o l l o w i n g t h e o r e m b y the m e t h o d s o f § 4 .

T H E O R E M 6.1. Assume that t^r) is a transcendency basis of K over R. Then there exist elements a^u. . . aⁿ b ^x , b ^r i n R with a^t<b^h such that for every r-tuple ( q , . . . , cr) of elements in S with a^c^bi there exists a place cp: K-+Svco o v e r R with (p(ti) = Cifor 1 ^ / ^ r .

Remark 6.2. A s s u m e that S has transcendency degree o v e r R. T h e n f o r given elements a ^u a ^r , b^u ...,b^r i n R w i t h aⁱ<bⁱ there exists a n r-tuple ( q , . . . , c^r) o f ele-

ments i n S w h i c h are a l g e b r a i c a l l y independent over R a n d s u c h that ^ for l . < / < r . Indeed, take a n y system ( w ^ . . . , u^r) o f a l g e b r a i c a l l y independent elements i n

S. R e p l a c i n g u^t by ± u^t o r ± uf¹ we m a y assume that a l l u^t are p o s i t i v e a n d not i n - finitely large o v e r R, i.e. that there exist elements dt i n R such that O^u^d^ T h e n the elements

c. = at + d -1ni( 6£- f l . )

have the r e q u i r e d p r o p e r t y . A n y place cp:K-+ Su oo over R w i t h cp{t^t) = c⁽ must be a n i n - j e c t i o n o f AT i n t o S. T h u s T h e o r e m 6.1 c o n t a i n s L a n g ' s t h e o r e m a b o u t e m b e d d i n g s o f

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real f u n c t i o n fields ( [ 1 5 ] , T h . 10). O u r p r o o f w i l l be very close t o the p r o o f i n [15], the m a i n difference b e i n g t h a t we replace the use o f S t u r m ' s t h e o r e m b y results o f section 4.

Remark 6.3. It is evident f r o m the arguments i n [1] that one m a y use t h e o r e m 6.1 to o b t a i n generalizations o f A r t i n ' s results o n definite functions i n [1] § 2 a n d § 4 , see e.g. L a n g ' s t h e o r e m 8 i n [15].

I n the p r o o f o f T h e o r e m 6.1 we m a y a l w a y s assume i n a d d i t i o n that the r-tuple ( cuc r) is a l g e b r a i c a l l y free over R. I n fact, f o r a given S one easily constructs a real closed field extension T o f S such that S is m a x i m a l l y a r c h i m e d i a n i n T a n d T c o n t a i n s a system ul9...9ur o f elements w h i c h are a l g e b r a i c a l l y independent a n d infinitely s m a l l over S (e. g. [ 1 5 ] , p . 389). I f ( cl9cr) is a n r-tuple i n S s u c h that there exists a h o m o m o r p h i s m \p:K^>T over R w i t h i/^(/ⁱ) = c^f + uⁱ⁹ then the c o m p o s i t i o n cp = Xoijj w i t h the c a n o n i c a l place l\T^>Suco o f T/S ( [ 1 5 ] , p . 380) has the values <p(r.) = c^r

W e m a k e the f o l l o w i n g arrangements for the p r o o f o f T h e o r e m 6 . 1 : 7>*(1) denotes the image o f the u n i t element o f W(K) u n d e r the transfer m a p Tr* f r o m W(K) t o

W(R(t^l9...⁹ t^r)). W e chose elements gi(t^l9...⁹ t^r)⁹...⁹ gⁿ(t^l9...⁹ t^r) i n R\t^l9...⁹ t^r~\ s u c h that 7>*(1) is represented by the space

(gi(t^l9...⁹t^r))±.»±(g^H(t^l9...⁹t^r))

over R(t^l9...⁹ t^r). W e further chose a fixed o r d e r i n g o f A ^ a n d denote b y o the signature o f R(t^l9...,t^r) c o r r e s p o n d i n g w i t h the r e s t r i c t i o n o f this o r d e r i n g t o R ( t^l⁹t^r) . F i n a l l y we denote by Q the signature o f S.

W e first p r o v e T h e o r e m 6.1 i n the case r = 1. W r i t e / instead o f t^v L e t

d1<d2< "< dN

be the elements o f R w h i c h o c c u r as roots o f (at least one o f ) the p o l y n o m i a l s g^t(t) i f these have a n y roots i n R. W e chose a <b i n R i n the f o l l o w i n g w a y : I f there are n o roots let a a n d b be a r b i t r a r y w i t h a<b. I f t<dt let b = dl a n d a a r b i t r a r y <dv I f d^t<t<d^{i + l} let a = dⁱ⁹ b = d^{i + 1}. I f d^N<t let a = d^N a n d b be a r b i t r a r y >a. A s s u m e c is a n element o f S w i t h a<c<b a n d c n o t i n R. B y (1.1) every g^t(t) decomposes i n Ä [ r ] i n a p r o d u c t o f a constant, some factors t - d^t a n d some factors o f type (t - ef + f² w i t h / # 0 . Therefore c l e a r l ygt( c ) ^ 0a n dQ( gi( c ) ) = a(gi(t))for 1 ^ / < / i ( c f [15] p . 386.) W e denote b y / the h o m o m o r p h i s m f r o m R(t) to S over R w i t h / ( / ) = c. T h e n

**e°X(Tr(l))=** t

^Q(^gi^(c))=

t

^e(g^t(t)) = a(Tr*(l)),

i = l i = l

N o w o c a n be extended t o K. T h u s b y L e m m a 4.1 we have ^ o ^ ( r r* ( l ) ) > 0 . B y L e m - m a 4.3 x^c ^a ⁿ be extended to a h o m o m o r p h i s m cp f r o m AT to S .

F r o m the thus p r o v e d special case r= 1 o f T h e o r e m 6.1 one easily obtains (see [ 1 5 ] , p. 387) a p r o o f f o r a l l r ^ 1 o f the f o l l o w i n g

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P R O P O S I T I O N 6.4. ( [ 1 5 ] , T h . 8.). Assume K is ordered and M is a finite set of non zero elements in K. Then there exists a place cp:K^> Ru oo over R such that for all x in M the value cp(x) is ^ oo, ^ 0 and of the same sign as x.

W e n o w p r o v e T h e o r e m 6.1 f o r a r b i t r a r y r ^ 1. B y P r o p o s i t i o n 6.4 there exists a place cp:K^> Rucc over R such that the values cp(tⁱ) = hⁱ are finite a n d s u c h that the values cp(g^j(tⁱ,..., t^r))=g^j(h^l,..., h^r) are n o n zero a n d have the same signs as the c o r - r e s p o n d i n g g j ( t^{l 9}t^r) . { R e c a l l that we have fixed a n o r d e r i n g o n K.} T h e n there exists a n element e > 0 i n R such that f o r a l l r-tuples ( q ,c r ) i n S w i t h h{ — 6 < ci< A|. + + e the values g^j(c^l,..., c^r) are n o n zero a n d o f the same signs as the c o r r e s p o n d i n g gj(c^l9...⁹ c^r) are n o n zero a n d o f the same signs as the c o r r e s p o n d i n g g^J(t^l,..., t^r).

A s s u m e that ( q ,c ^r) is such a n r-tuple w h i c h is i n a d d i t i o n a l g e b r a i c a l l y free over R.

D e n o t e b y x the h o m o m o r p h i s m f r o m R(tl9..., tr) to S over R w h i c h m a p s tt t o ci f o r 1 ^ / ^ r . A s i n the case r = 1 we o b t a i n

o{Tr*(\)) = Qo^U(Tr*{l))

a n d we see again b y L e m m a 4.1 a n d 4.3 that / c a n be extended to a h o m o m o r p h i s m cp f r o m Kio S. T h e o r e m 6.1 is p r o v e d .

C e r t a i n l y there are other relevant questions a b o u t real places o n f u n c t i o n fields w h i c h c a n be treated b y the m e t h o d s used here. I hope to c o m e b a c k t o this subject in the near future.

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Mathematisches Institut der Universität des Saarlandes Deutschland

Received March 4, 1972