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Valued fields with a universal embeddingproperty and ηα -structuresBecker Thomas aa Fakultät für Mathematik und Informatik , Universität Passau , Postfach 2540,Passau, D-8390, West GermanyPublished online: 27 Jun 2007.

To cite this article: Becker Thomas (1990) Valued fields with a universal embedding property and ηα -structures, Communications in Algebra, 18:10, 3565-3576, DOI: 10.1080/00927879008824091

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COMMUNICATIONS IN ALGEBRA, 18(10), 3565-3576 (1990)

VALUED FIELDS WITH A UNIVERSAL EMBEDDING PROPERTY AND q,-STRUCTURES

Thomas Becker

Fakultit fCr Mathematik und Informatak, Universitit Passau Postfach 2540, 0 -8990 Passau, West Germany

Abst rac t We prove that if the continuum hypothesis is assumed and K is a regular

uncountable cardinal, then there is a valued field of cardinality n and equicharacteristic p

into which every valued field of cardinality 5 n and equicharacteristic p can be embedded.

The same is true for a certain class of ordered valued fields. We prove a general existence

theorem for r),-structures which provides a simple uniform proof for the existence of q,-

sets, 7,-groups, and 7,-fields.

0. Introduction An ordered set T is called an 77,-set if it has the following property: whenever H, I( C T, H < K , and (HI, (K( < N,, then there exists a E T with H < a < K ([Ill VI $8). In [Ill, [15], and [8], the following is proved.

(i) If T is an 7,-set, then every ordered set S with IS1 < N, can be embedded in T.

(ii) Any two 7,-sets of cardinality N, are isomorphic.

(iii) If N, is regular and sup2N@ 5 N,, @<a

then there exists an 7,-set of cardinality N,.

An 7,-group (77,-field) is an ordered abelian group (ordered field) whose underlying ordered set is an 77,-set. In [7], [I], and [2] it is proved that (i)- (iii) continue to hold if we replace "7,-set" by "divisible r],-group " ("real closed 7,-field") and require a -> 0. In section 1 of this paper, we prove

Copyright O 1990 by Marcel Dekker, Inc.

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the existence of certain valued fields with a universal embedding property similar to (i) above.

We write (K, r, v) for a valued field K with non-archimedean valuation v and value group I?. Recall that ( K , I?, v) is said to have equicharacteristic p (p prime or 0) if both Ii' and the residue class field of (K, r , v ) have characteristic p. ( K l , r l , v l ) E (K2,1?2,v2) means Kl K2, I?* r2, and v2EKl = vl. An embedding (4, h) : (Kl, rl, vl) - (K2, r2, v2) consists of embeddings 4 : K1 - h; and h : I?l - l72 such that vl = v2 o 4. If F is a field and r an ordered abelian group, then we denote by F ( ( r ) ) the field of formal Laurent series over r with coefficients in F, and if in addition, a > 0 is an ordinal, then by F((I')),, we mean the subfield of F ( ( r ) ) that consists of all f E F((I?)) with Jsupport(f)J < N,. If K is any subfield of F( ( r ) ) , then (K, I?, v ) will always mean h' with the natural valuation given by v(f) = inf(support(f)). We will prove the following theorem.

Theorem 1 Let N, be a regular cardinal such tha t a > 0 and

F a n algebraically closed field wi th )FI = N, and char(F) = p (p prime o r O), I? a n 71,-group wi th IF( = N,, and K = F((I?)),. T h e n IKl = H,, and every diagram

( $ 7 k) 1 1 1 - (Kz, rz, v2)

where lKll < N,, IKzJ 5 N,, and (K~,I'z,vz) i s o f equicharacteristic p can be completed as follows.

In particular, every valued field o f cardinality 5 N, and equ~characterzst ic p can be embedded i n (I<, I?, v ) .

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It should be noted that the strength of the above theorem lies in the fact that the cardinality of K itself is no more than N,. One could ob- tain the existence of valued fields with the same universal property using maximal immediate extensions, but these would in general have higher car- dinalities. Our theorem could also be interpreted as stating the existence of "a- in jec t i ve" valued fields, where a-injectivity has the obvious meaning.

In section 2, we prove a similar theorem for certain valued fields (I<, I?, v) where K is ordered. ( K , I?, v) is called an ordered valued field if K is an ordered field whose order and valuation satisfy the following compatibility condition:

These have been investigated under various aspects in [13], [5], [4], and [3]. In the context of ordered valued fields, inclusions will be understood to be inclusions of ordered fields too, and embeddings will be understood to be order-preserving. If F is an ordered field, r an ordered abelian group, and K a subfield of F((F)), then ( K , r, v) will always be K with its natural val- uation as before, and K will be understood to be ordered lexicographically, such that f > 0 iff f (v( f)) > 0. It is easy to see that (K, r, v) is then an ordered valued field. We will prove the following theorem.

Theorem 2 L e t N, be a regular cardinal w i t h a > 0 a n d

sup2 '~ 5 N,, @<a

F a real closed q,-field of cardinality N,, F a n .r),-group of cardinal i ty N,, K = F((r)) , . T h e n (K, I?, v) i s a n ordered valued field w i t h Ih'l = N,, a n d every d iagram

( $ 9 k) 1 1 1 - (K2, r2, v2)

where IK1l < H a , IK2l 5 N,, a n d (IG, rl, vl) and (Kz, r2, v2) are ordered valued fields c a n be completed a8 follows.

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In particular, every ordered valued field of cardinality 5 N, can be ev,bedded zn (A', I', v).

We wish to point out an important difference between the situation for ordered fields and for ordered valued fields. For o > 0, A real closed field whose underlying set is an 77,-set automatically has the universal embedding property for ordered fields of cardinality 5 N,. This is not true for ordered valued fields. If we perform the same construction as in the above theorem but take for F the real numbers instead of an 77,-field, then by [2], I< is an %-field. However, it is clear that (K, r, v) does not have the universal embedding property for ordered valued fields of cardinality 5 N,: its residue class field is too small. Here, we have to have an ordered valued field whose residue class field is also an 11,-field.

The above mentioned proofs for the existence of 77,-sets, 77,-groups, and 11,-fields are very different in nature. In section 4 of this paper, we give a simple proof of a general existence theorem of 7,-structures which covers these three cases. It does not provide a very explicit description of va- structures, but its proof sheds sonie light on the reasons for their existence. We call an ordered algebraic structure A an 11,-structure if its underlying set A is an 17,-set. By an abuse of notation, we will write < for the order on any ordered algebraic structure A. When we write A C B, it is understood that the inclusion preserves order and algebraic structure. We will prove the following theorem.

Theorem 3 Let N, be a regular cardinal such tha t

0 # a set of ordered algebraic structures, each of cardinality 5 N,. A s s u m e tha t is closed u n d e r un ions of infinite ascending chains of length 5 N,, and tha t for all A E Em, H, h' C_ A such tha t H U K = A and H < K , there exists B E Ca with A 23 and b E B with H < b < Ii'. T h e n contains a n 0,-structure.

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The assumptions of the theorem are sat isfief d if we take for Ca the set of all ordered sets (divisible ordered abelian groups, real closed fields) of cardinality 5 N,.

The author is indebted to L. Fuchs for suggesting the investigation of valued fields with universal embedding properties, and for helpful conver- sations.

1. Valued fields The following lemma contains the main argument of the proof of theorem 1. If (K, F, v) is a valued field, we write res(K) for its residue class field.

Lemma 1.1 Let (I<,F,v) be as i n theorem 1, L an algebraically closed subfield of K with ILI < N,, L(x) a simple transcendental valued extension of L with valuation w and value group F' E I? such that wb L = V I L. Then there exists a n embedding 4 : L(x) --+ K over L with w = v o 4.

Proof Since L is algebraically closed, any valuation on a simple transcen- dental extension L(y) of L extending a given valuation on L is completely determined by its values on the linear polynomials x - g, where g E L. Hence it suffices to find f E K with f 4 L (and hence f transcendental over L) and v( f - g) = w(x - g) for all g E L. We distinguish between three cases.

Case 1: v(L) # I". Then w(x -g) 6 v(L) for at least one g E L, and we may as we11 assume that W(X) = y v(L). Then w(x -g) = min(y, v(g)) for all g E L. Let f be the element of K that satisfies f (y) = 1 and f (y') = 0

for all y' E I' \ (7). Then f 4 L and v(f - g) = min(y, v(g)) = w(x - g) for all g E L.

Case 2: v(L) = I?'. Then there is a natural embedding x : res(L) - res(L(x)). There are two subcases which need to be treated separately.

Case 2a: x is not onto. By our assumption v(L) = I", we can find g E L with w(gx) = 0, hence we may assume that w(x) = 0. Furthermore, w(x - g) = 0 for all g E L with v(g) = 0 since the residue class of x - g is not in x(res(L)), and we get w(x - g) = min(0, v(g)) for all g E L. Pick a E F \ {g(O) I g E L, v(g) = 01, and define f E K by setting f(0) = a, f (7) = 0 for y # 0. Then clearly f E K \ L, and v( f - g) = min(0, v(g)) = w(x - g) for all g E L.

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Case Zb: x is onto, i.e. (L(x),rr , w) is a n immedia te extension of (L, v(L), V E L). By [12], theorem 1, x is the limit of a pseudo-convergent sequence in (L, v(L), U P L) which does not have a limit in L. Since (LI < N,, this sequence has a limit f E I<, and by [12], theorems 2 and 3 and the fact that L is algebraically closed, f has the desired properties. (The point here is that the pseudo-convergent sequence cannot be of algebraic type in the sense of 1121: if it were, it would have to have a limit in L. Hence theorem 2 of [12] applies.)

We are now in a position to prove theorem 1.

Proof of theorem 1 It follows easily from our assumption on N, that IKI = N,. Now let (Kl, Fl, vl) and (K2, r2, v2) be as in the theorem. We may assume that K2 is algebraically closed, and that 4 and + are inclusions. By [I], theorem A , we can embed r2 into F over rl, and we may thus assume that rl c F2 C F. Let { x u I v < wp } be a transcendence base of K2 over K1. We define an ascending chain {Lv}v<wp of subfields of h'2 as follows. Let Lo be the algebraic closure of K1 in Iq2. For limit ordinals A, we set

LA = U L,, v<X

and for v = p + 1 let L, be the algebraic closure of L,(x,) in K2. It is clear that

K 2 = u L,. u<wp

To get the desired embedding of (K2, r2, v2) into (I<, I?, v) over (Kl, rl, vl), it now suffices to find embeddings 4, : L, - IT over K1 such that 4, extends 4, for all 0 5 p 5 v < wp and v2PL, = v o 4, for all 0 5 v < wp. It follows easily from [14], theorem 1, that K is algebraically closed. Hence there is an embedding 4; : Lo - K over K1. By [6] (14.3) and (14.7), there is a in the Galois group of Lo over I<l such that v2 1 Lo = v o 4; o a, and we set +o = $boa. For simplicity, we identify +,(L,) with L, whenever 4, has been defined. We can then take = idL, for all limit ordinals X < wp. Finally, if v = p + 1, then lemma 1.1 provides an embedding 4; : L,(x,) - h' with v2PL,(x,) = v o &,, and we obtain 4, as in the case v = 0 by extending 4; to L, and then composing it with a suitable element of the Galois group of L, over L,(x,). 0

2. Ordered valued Aelds For the proof of theorem 2, we need several preparatory lemmas.

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Lemma 2.1 L e t ( K , r , v ) be a n ordered valued field, a , b E K w i t h a , b 2 0 o r a , b 5 0. T h e n v ( a + b) = m i n ( v ( a ) , v (b ) ) .

Proof The claim is trivial if one or both of a , b are 0. We have c < c + 1 for all c E K , hence v ( c + 1 ) 5 v ( c ) for all 0 5 c E I{. Now if 0 < a , b, then

and similarly v ( a + b) 5 v(b) . If a , b < 0, then

Lemma 2.2 L e t ( I i , I?, v ) be a n ordered valued field, K = L ( x ) w i t h L real closed a n d x t ranscenden ta l o v e r L . T h e n v is complete ly d e t e r m i n e d by v l L and t h e values { v ( x - c ) 1 c E L ).

Proof Every polynomial f ( x ) in L [ x ] can be written in the form

m n

f ( x ) = c n ( x - ci) n [ ( x - dj)' + k j ] i=l j= 1

with c , c ; , d j , kj E L and Lj > 0 (1 5 i 5 m, 1 5 j 5 n). The claim now follows from lemma 2.1. 0

Lemma 2.3 L e t ( K , F , v ) be a n ordered valued field, a , c l , c2 E IC wi th v ( a - c 2 ) < v ( a - c l ) . T h e n a - c2 and c2 - cl have opposi te s igns .

Proof It is easy to see that a # c2 and cl # c2. We have

We obtain

It now follows from 2.1 that a - c2 and c2 - cl must have opposite signs.

Lemma 2.4 Let ( K , r , v ) be a n ordered valued field w i t h K real closed. I f w i s a n e x t e n s i o n o f v t o K ( i ) , t h e n ( K ( i ) , r , w ) i s n o t a n i m m e d i a t e e x t e n s i o n o f ( K , r, v ) .

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Proof Clearly, X2 + 1 has a zero in res(K(i)). We show that this is not the case in res(K). Assume for a contradiction that v(a2 + 1) > 0 for some a E Ii'with v(a) 1 0 . By 2.1, v(a2+1) = min(v(a2),0) = 0, a contradiction. 0

The following lemma is the analogue of 1.1.

Lemma 2.5 Let (K,I',v) be as in theorem 2, L a real closed subfield of I< with IKI < N,, L(x) a n ordered simple transcendental extension of L with valuation w and value group r' E r such tha t (L, v(L), vlL) c (L(x), I?', w) as ordered valued fields. T h e n there exists a n embedding 4 : L(x) ---t Ii' of ordered fields with w = v o 4.

Proof By 2.2 and the fact that the order of a simple transcendental exten- sion R(y) of a real closed field R is determined by the order of R u {y) (see e.g. [9] 13.12), it suffices to find f E I< with f $ L, v(f - g) = w(z - g)

for all g E L, and f - g > 0 in the order of K iff x - g > 0 in the order of L(x) for all g E L. The proof parallels the proof of lemma 1 . l .

Case 1: v(L) # I?'. Then w(x - g) $! v(L) for some g E L, and we may assume w(x) = y @ v(L) and x > 0. Then w(x - g) = min(y, v(g)) for all g E L, and x < g iff v(g) < y for all 0 5 g E L. Let f be the element of I< that satisfies f (y) = 1 and f(yl) = 0 for all y' # y. Then f $! L, v( f - g) = min(y, v(g)) = w(x - g) for all g E L, and f < g iff v(g) < y iff x < g for all 0 5 g E L. It is not hard to see that the latter equivalence must then hold for all g E L.

Case 2: v(L) = I?'. Then there is a natural embedding ,y : res(L) - res(L(x)). There are two subcases which need to be treated separately.

Case 2a: ,y is no t onto. By our assumption v(L) = I", we can find g E L with w(gx) = 0, hence we may assume that w(x) = 0 and x > 0. Furthermore, w(x - g) = 0 for all g E L with v(g) = 0 since the residue class of x - g is not in x(res(L)), and we get w(x - g) = min(0, v(g)) for all g E L. As in the proof of lemma 1.1, we will define f by setting f (y) = 0 for all y # 0 and f(0) = a E F \ {g(O) I g E L, v(g) = 01, but this time with an additional requirement on a. Obviously, D

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Since F is an 77,-field and ILI < N,, we can choose our a strictly between these two sets. Then f E K \ L and v( f - g) = min(0, v(g)) = w(x - g) for all g E L. To show that f - g > 0 iff x - g > 0 for all g E L, it suffices to prove this for 0 5 g E L since x > 0. We consider two cases. If v(g) # 0, theng < f iff0 < v(g) iff g < x. Ifv(g) = 0, theng < f iffg(0) < f(0) = a iff g < x.

Case 2b: x is onto, i.e. (L(x),rl, w) is a n i m m e d i a t e ex tens ion of

(L, v(L), V P L). Then x is the limit of a pseudo-convergent sequence in (L, v(L), vEL) which does not have a limit in L, and we take for f any limit of this sequence in K . By 2.4 and [12], theorems 2 and 3, v( f -g) = w(x -g) forallgE L. Toseethat f - g > O i f f x - g > O , l e t g E L. I t i snotedinthe

J proof of [12], theorem 1, that there must beg' E L with w(x-g) < w(x-g'). Then by lemma 2.3, x - g and g' - g have the same sign. By the same argument, f - g and g' - g have the same sign.

It was proved in [13] that if (I(,r,v) is any ordered valued field, I" the divisible closure of I?, and L the real closure of K , then there exists a unaque valuation w on L such that (L, I", w) is an ordered valued field and (K, r, v) (L, I?', w). The proof of theorem 2 is now similar to the one of theorem 1.

Proof of theorem 2 It follows easily from our assumption on N, that IKI = N,. Now let (K1, rl, vl) and (K2, r2, v2) be as in the theorem. We may assume that K2 is real closed, and that 4 and 1C, are inclusions. By [I], theorem A, we can embed r2 into I' over rl, and we may thus assume that rl c r2 r . Let {x, 1 Y < w p ) be a transcendence base of K2 over Kl. We define an ascending chain {L,),<, of subfields of K2 as follows. Let Lo be the real closure of K1 in K2. For limit ordinals A , we set

LA = U L,,, ,<A

and for v = p + 1 let L, be the real closure of L,(x,) in K2. Then

and it suffices to find order preserving embeddings 4, : L, - K over Kl such that 4, extends 4, for all 0 5 p 5 v < wp and vzlLv = v o 4, for all 0 5 v < wp. It follows easily from [14], theorem 1, that K is real closed. Hence there is an embedding &, : Lo + K over Kl. By the remark preceding the proof, this embedding must automatically satisfy v2PLo = v o q50. Again, we identify $,(L,) with L, whenever 4, has been defined and take = idL, for all limit ordinals X < wo. Finally, if Y = p+l ,

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then lemma 2.5 with v2E L,(z,) embedding 4, :

provides an embedding 4; : L,(x,) - K of ordered fields = v o @,, and as in the case v = 0, this extends to an L, - K with the desired properties.

3. 7,-Structures We produce the 7,-structure whose existence is claimed in theorem 3 by two nested ascending chain constructions.

Lemma 3.1 Let N, and C" be as an theorem 3, A E Ca. Then there exists d C d E Ca such that whenever H, K C A wath [HI, IKI < H a and H < K , then there is a E a with H < a < K.

Proof By our assumption on N,, there exist at most N, many subsets of A of cardinality < N,. Let { (H,, K,) ( v < wp ) ( P 5 cr) be an enumeration of all pairs (H, K ) where H, K C A, lH 1, I K I < N,, and H < K. Define an ascending chain {A,},<wp of elements of Ca as follows. Set A0 = A, and

for limit ordinals A. If v = p + 1, consider the sets

H = { a € A, l a 5 h forsome h E H,)

and K = { a € A , I a > h f o r a l l h ~ H , } .

Then H < K and H U K = A,. Hence we can find A, E A, E Cu such that there is a E A, with H < a < K. Then obviously H, < a < K, too. It is clear that

d = u EX, v< wg

has the desired property. The proof of theorem 3 is now as follows.

Proof of theorem 3 We define an ascending chain {A,),<,, as follows. Take for any element of Xu. Set

for limit ordinals A, and A, = d, if v = p + 1. We claim that

A = U A, u<w4

is an 7,-structure. Indeed, if H, K C A with /HI, JKI < N, and H < K,

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[14] S. McLane, The universality of formal power series fields, Bull. AMS 45 (1939), 888-S90

[15] W. Sierpinski, Sur une proprikttr des ensembles ordonnks, Fund. Math. 36 (1949), 56-67

Received: August 1989

Revised: January 1990

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