Henselian Valued Stable Fields

206, 344]369 Ž1998. JA977396

JOURNAL OF ALGEBRA ARTICLE NO.

Henselian Valued Stable Fields I. D. Chipchakov* Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonche¨ Str., bl. 8, 1113, Sofia, Bulgaria Communicated by Walter Feit Received November 10, 1997

INTRODUCTION A field E is said to be stable, if the Schur index indŽ D . of each central division E-algebra D of finite dimension is equal to the exponent expŽ D ., i.e., the order of the similarity class w D x as an element of the Brauer group BrŽ E .; we say that E is stable closed, if its finite extensions are stable fields. The class of stable closed fields is considerably smaller than the class of stable fields Žsee Proposition 4.1 and Corollaries 2.6, 4.4, and 4.5.. Both classes are naturally singled out by the well-known general relations between indices and exponents of central simple algebras Žcf. wP, Sects. 14.4 and 19.6x.. Index-exponent relations are subject to additional restrictions when the centres of these algebras are global fields or other fields often used in algebra, number theory, and algebraic geometry Žcf. wP, Sects. 17.10 and 18.6; Ar, Theorem Ž1.1.; L 2 , Theorems 5, 6, and 12; MS, Ž16.8. ] Ž16.10.x.. The corresponding results have been obtained as consequences of arithmetic, diophantine, topological, or other specific properties of the centres. In particular, the stability of global fields and local fields follows from the description of their Brauer groups by class field theory Žcf. wWe, Chap. XII, Sect. 2, and Chap. XIII, Sect. 6x.. We refer the reader to wAr, FSax, for examples of stable fields given by commutative algebra and the theory of algebraic surfaces. It is not known whether the mentioned results can be proved and interpreted in a unified manner. This raises the interest in the systematic study of stable fields from suitably chosen special classes. * Partially supported by the Bulgarian Foundation ‘‘Scientific Research.’’ 344 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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As a step in this direction, this paper deals with the problem of characterizing stable fields with Henselian valuations by the algebraic properties of their value groups and residue fields. The obtained results give a full answer to the discussed question for several important classes of the considered valued fields, including the classes of almost perfect equicharacteristic fields, discrete valued fields with perfect residue fields, and iterated Laurent formal power series fields in n G 2 indeterminates. They also show that the only obstruction to the general solution of the posed problem is laid by the relations between Schur indices and exponents of central division algebras of q-primary dimensions over Henselian valued fields with residue fields of characteristic q ) 0. The paper gives a possibility of studying stable fields by applying various constructive methods. This enables one to understand better a number of properties of these fields, e.g., the behaviour of their Galois cohomological invariants and the structure of their Brauer groups Žsee Corollary 4.8 and wCh 3 , Ch 4 x.. Throughout this paper, simple algebras are assumed to be associative with a unit and of finite dimensions over their centres. As usual, Brauer groups of fields are considered to be additively presented, Galois groups are regarded as profinite with respect to the Krull topology, and homomorphisms of profinite groups are supposed to be continuous. Our basic terminology and notations concerning valuation theory, simple algebras, Brauer groups, field extensions, Galois theory, profinite groups, and Galois cohomology are standard Žas those used, for example, in wE, JW, TY, P, L 1 , Se, Kx.. For convenience of the reader, we recall in Section 1 a part of the vocabulary of valuation theory, together with some preliminaries needed for the further discussion. The main results of the paper are presented in Sections 2 and 3: Theorem 2.1 describes the basic relations between value groups and residue fields of stable fields with Henselian valuations, and Theorem 3.1 gives a general sufficient condition for stability of Henselian valued fields. In Section 4, we discuss consequences of the main results. They show that the stability of a Henselian valued field allows much greater diversity in its structure than the property of being stable closed.

1. PRELIMINARIES AND THE FIRST NECESSARY CONDITION FOR STABILITY OF HENSELIAN VALUED FIELDS Let K be a field with a Krull valuation ¨ , and let ¨ Ž K ., OK , and Kˆ be the value group, the valuation ring, and the residue field of Ž K, ¨ ., respectively. We say that ¨ Ž K . is a totally indivisible group, if it is p-indivisible Ži.e., ¨ Ž K . / p.¨ Ž K .., for every prime number p. The valuation ¨ is called Henselian, if ¨ Ž K . /  04 and each of the following

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equivalent conditions holds: Ž1.1. Ži. For any monic polynomial f g OK w X x, if the reduction

ˆ then f has a simple root a in OK such fˆg Kˆw X x has a simple root a ˜ g K, that a ˆ s a; ˜

Žii. ¨ extends to a uniquely determined valuation on each algebraic extension of K; Žiii. Every division K-algebra of finite dimension has a unique valuation extending ¨ . For a proof of the equivalence of the conditions in Ž1.1., we refer the reader to wR, Er, Wx. It is clear from Ž1.1.Ži. that K has the following properties when ¨ is Henselian:

ˆ Then: Ž1.2. Let p be a prime number not equal to char K. Ži. 1 q a is a pth power of an element of OK whenever ¨ Ž a . ) 0; Žii. The multiplicative group K *r K * p is isomorphic to the direct sum Kˆ*r Kˆ* p [ ¨ Ž K .rp.¨ Ž K .. Assume as above that Ž K, ¨ . is a Henselian valued field and let D be a division K-algebra of finite dimension w D: K x. The main algebraic strucˆ and value group tures associated with D are its residue division ring D ¨ Ž D . Žor ¨ D Ž D .. with respect to the valuation ¨ D of D extending ¨ . It is ˆ is a K-algebra ˆ ˆ Kˆx dividing w D: K x, well known that D of dimension w D: ¨ Ž D . is an abelian group, and ¨ Ž K . is a subgroup of ¨ Ž D . of index ˆ Kˆxy1 ; more precisely, by the Ostrowski]Draxl theoeŽ DrK . F w D: K x.w D: rem, we have

ˆ Kˆx.eŽ DrK ..dŽ DrK ., for some natural numŽ1.3. Ži. w D: K x s w D: ber dŽ DrK . Žcalled a defect of D over K .; Žii. If dŽ DrK . / 1, then char Kˆ s q ) 0 and dŽ DrK . is a power of q. As shown in wTYx, this leads to a complete description of the relations between defects and Schur indices of central division algebras over Henselian valued fields. The K-algebra D is called defectless Žwith respect to ¨ . if dŽ DrK . s 1, and it is called immediate if dŽ DrK . s w D: K x. For example, D is defectless over K provided that w D: K x is not divisible by ˆ this condition is fulfilled, if D is central over K and K is a perfect char K; field with char K s char Kˆ Žcf. wP, Sect. 14.4; A, Chap. VII, Theorem 22x.. The defectlessness of central division K-algebras is also guaranteed when Ž K, ¨ . is a Henselian discrete valued field or an n-discretely valued field, e.g., an iterated Laurent formal power series field in n indeterminates wTY, Propositions 2.1 and 2.2x.

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The K-algebra D is said to be totally ramified, if eŽ DrK . s w D: K x. An algebraic extension K 1 of K is called totally ramified, if eŽ K 1X rK . s w K 1X : K x, for every finite extension K 1X of K in K 1. We say that D is inertial ˆ Kˆx and the centre ZŽ Dˆ. of Dˆ is a separable over K, if w D: K x s w D: ˆ extension of K. By Theorem 2.8Ža. of wJWx, for every finite dimensional ˆ ˆ there exists an inertial division K-algebra S˜ with ZŽ S˜. separable over K, ˆ ˆ ˜ The K-algebra S is division K-algebra S such that S is K-isomorphic to S. ˜ uniquely determined by S up to an isomorphism. This algebra is called an inertial lift of S˜ over K. The main necessary condition for stability of Henselian valued fields obtained in wCh 4 x can be stated as follows: PROPOSITION 1.1. Let K be a stable field with a Henselian ¨ aluation ¨ . Then the residue field Kˆ is also stable. Moreo¨ er, if ¨ Ž K . is a p-indi¨ isible group for some prime number p, then e¨ ery cyclic p-extension Eˆ of Kˆ embeds ˆ as a subalgebra in each central di¨ ision K-algebra of index di¨ isible by the ˆ Kˆx. degree w E: The description of the algebraic properties of stable fields with Henselian valuations is simplified by the notions of a Z p-extension, an almost perfect field, a quasi-local field, and a p-group of Demushkin type, and also, by some notions used in the theory of ordered fields. Let us note that a field M is said to be formally real, if y1 is not presentable over M as a finite sum of squares; M is called a nonreal field, otherwise. We say that M is Pythagorean, if it is formally real and the element a2 q b 2 is a square in M, for each pair Ž a, b . g M = M. By a Z p-extension, we mean a Galois extension with a Galois group isomorphic to the additive group Z p of p-adic integers. A field E is said to be almost perfect, if every finite extension of E is simple, i.e., generated by a primitive element. This condition can be restated by saying that either char E s 0 or char E s q ) 0 and the degree of E over the subfield E q s  a q : a g E4 is equal to 1 or q Žcf. wCh 4 , Ž4.1.x.. The described alternative is valid, for example, when E is a rational function field or a formal Laurent power series field in one ˜ in these cases, E is perfect if and indeterminate over a perfect field E; only if char E s 0. A field F is called quasi-local, if the following statement is true, for every finite extension F1rF: If F1X rF1 is a cyclic extension and D is a central division F1-algebra of index divisible by the degree w F1X : F1 x, then F1X embeds in D as an F1-subalgebra. It has been proved in wCh 4 x that the class of quasi-local fields contains the residue fields of Henselian valued stable closed fields with totally indivisible value groups. Note also that a Henselian discrete valued field Ž K, ¨ . lies in this class if and only if Kˆ is perfect and the absolute Galois group of Kˆ is metabelian of cohomological dimension F 1 wCh 3 , Corollary

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2.5x. This generalizes the well-known fact that local fields are quasi-local Žsee also wCF, Chap. VI, Appendix to Sect. 1x.. Other examples of quasi-local fields and more information about them can be found in wCh 2 , Ch 3 , Ch 4 x. An infinite pro-p-group P is said to be a p-group of Demushkin type, if the homomorphism wa : H 1 Ž P, Fp . ª H 2 Ž P, Fp . mapping each b g H 1 Ž P, Fp . into the cup-product a j b is surjective, for every nonzero element a g H 1 Ž P, Fp . Žwhere H j Ž P, Fp . is the jth cohomology group of P with coefficients in the prime field Fp of characteristic p, see wSe, Chap. I, 2.2; K, 3.9x.. This condition is satisfied when P is a free pro-p-group Žcf. wSe, Chap. I, 4.1 and 4.2x. or a Demushkin pro-p-group of finite or countable rank Žcf. wD, Lab 1 , Lab 2 , MWx.. It also holds when P is ˆ isomorphic to the Galois group of the maximal p-extension KˆŽ p .rK, Ž . provided that K, ¨ is a Henselian valued stable nonreal field such that ˆ and K contains a primitive pth root of unity ¨ Ž K . / p.¨ Ž K ., KˆŽ p . / K, Žsee Proposition 1.1 and wCh 4 , Lemma 3.9; Se, Chap. II, Proposition 3x.. Groups of Demushkin type and quasi-local nonreal fields are related as follows Žcf. wCh 4 , Theorem 4.1 and Lemmas 3.9 and 4.5x.: PROPOSITION 1.2. Let E be a nonreal field, Es e p a separable closure of E, and GE [ GŽ Es e prE . the absolute Galois group of E. Ži. If E is a quasi-local field and p is a prime number for which the cohomological p-dimension cd p Ž GE . is ) 0, then each open subgroup of a Sylow pro-p-subgroup of GE is a p-group of Demushkin type; Žii. If E is perfect and the Sylow pro-p-subgroups of GE are of Demushkin type together with their open subgroups, whene¨ er cd p Ž GE . ) 0, then E is quasi-local. In addition to Proposition 1.2, the results of wCh 3 , Sect. 3x contain an essentially complete characterization of quasi-local formally real fields by their absolute Galois groups. They also give a classification of the profinite groups realizable as such Galois groups.

2. RELATIONS BETWEEN RESIDUE FIELDS AND VALUE GROUPS OF HENSELIAN VALUED STABLE FIELDS Proposition 1.1 indicates that the study of a Henselian valued stable field Ž K, ¨ . should concentrate not only on the general properties of the ˆ but also on the specific relations between Kˆ and the residue field K, quotient group ¨ Ž K .rp.¨ Ž K . arising when p is a prime number for which ¨ Ž K . / p.¨ Ž K .. The main result of this section proves that ¨ Ž K .rp.¨ Ž K .

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is a cyclic group, provided that either GŽ KˆŽ p .rKˆ. is a pro-p-group of rank G 2 or Kˆ is formally real and p s 2. In addition, it turns out that ¨ Ž K .rp.¨ Ž K . is of order F p 3 whenever Kˆ contains a primitive pth root of unity: THEOREM 2.1. Let K be a stable field with a Henselian ¨ aluation ¨ and let p be a prime number. Then K, ¨ , the rank t Ž p . of the abelian p-group ˆ and the maximal p-extension KˆŽ p .rKˆ ¨ Ž K .rp.¨ Ž K ., the residue field K, satisfy the following conditions: Ži. If K contains a primiti¨ e pth root of unity or char K s p, then t Ž p . F 3; the equality t Ž p . s 3 is possible only in the case of char K / p. ˆ ˆ p is a finite extension of Moreo¨ er, if char Kˆ s p and t Ž p . F 2, then KrK d degree p , for some integer d F 2 y t Ž p .. ˆ then Kˆ does not contain a On the other hand, if t Ž p . G 3 and KˆŽ p . / K, primiti¨ e pth root of unity. Žii. If t Ž p . s 3 and K contains a primiti¨ e pth root of unity, then it contains a primiti¨ e root of unity of each p-power degree and the Galois group GŽ K Ž p .rK . is isomorphic to the additi¨ e group Z p3 of 3-tuples of p-adic integers, pro¨ ided with the natural topology. Moreo¨ er, if char Kˆ s p, then Kˆ is a perfect field and ¨ Žp . ) t.¨ Ž p ., for all rational integers t and e¨ ery p g K *, such that ¨ Žp . ) 0 and ¨ Žp . f p.¨ Ž K .. Žiii.

ˆ If t Ž p . G 2, then KˆŽ p .rKˆ is a Z p-extension unless KˆŽ p . s K.

Živ. If Kˆ is formally real and ¨ Ž K . / 2.¨ Ž K ., then Kˆ is Pythagorean, KˆŽ2.rKˆ is a quadratic extension, and t Ž2. s 1. Remark 2.2. Note that if Ž L, w . is a valued finite extension of a Henselian valued field Ž K, ¨ ., then the value group w Ž L. is totally indivisible if and only if so is ¨ Ž K .. More precisely, it follows from Ž1.3. that the p-groups ¨ Ž K .rp.¨ Ž K . and w Ž L.rp.w Ž L. are of one and the same rank t Ž p . F q`, for each prime number p. Furthermore, if ¨ Ž K . / p.¨ Ž K ., the degree w L: K x is not divisible by p and S is a subset of K *, then the co-set  ¨ Ž l. q p.¨ Ž K .: l g S4 is a minimal set of generators of the quotient group ¨ Ž K .rp.¨ Ž K . if and only if  w Ž l. q p.w Ž L.: l g S4 is such a set for w Ž L.rp.w Ž L.. Therefore, Proposition 1.2 and Theorem 2.1 can easily be applied to the study of the basic types of Henselian valued stable closed fields. Proof. We show that the conditions of Theorem 2.1 are necessary to exclude the existence of any central division algebra over K which is a tensor product of two cyclic K-algebras of index p. Our argument is grounded on the following lemma.

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LEMMA 2.3. Let E1 and E2 be cyclic extensions of a field E of prime degree p. Denote by Di the cyclic E-algebras Ž Ei , si , bi ., for some elements bi g E*, a fixed E-automorphism si of the field Ei of order p, and i s 1, 2. Then D 1 mE D 2 is a central di¨ ision E-algebra if and only if D9 [ D 1 mE E2 py1 is a di¨ ision algebra, such that the product d s Ł is0 s 2X i Ž d . is not equal to X b 2 , for any d g D9, s 2 being the unique E-automorphism of D9 inducing s 2 on E2 and acting on D 1 as the identity. Lemma 2.3 is obtained in a straightforward way from Lemma 1.2 and wA, Chap. XI, Theorems 11 and 12x. Suppose now that the field K has an extension K 3 representable as a compositum of two distinct cyclic extensions K 1 and K 2 of K of degree p, and let D 1 , D 2 , D9, s 1 , s 2 , s 2X , b1 , and b 2 be defined as in Lemma 2.3, for E s K, E1 s K 1 , and E2 s K 2 . It is known Žcf. wP, Sect. 13.4, Corollaryx or wCh 1 , Lemma 3.5x. that D9 is a division algebra if and only if K 2 does not embed in D 1 as a K-subalgebra. When D9 is such an algebra, its unique valuation ¨ 9 extending ¨ has the property that ¨ 9Ž s 2X Ž d .. s ¨ 9Ž d . and the value of the product d defined in Lemma 2.3 is an element of p.¨ 9Ž D9., for every d g D9. Note also that s 2X induces in this case an automorphism ˆ of the residue division K-algebra of Ž D9, ¨ 9.. Considering K 1 , K 2 , and K 3 with their valuation extending ¨ , one obtains from formula Ž1.3., Lemma 2.3, and these observations that D 1 mK D 2 is a division K-algebra whenever the subgroup of ¨ Ž K .rp.¨ Ž K . generated by the co-classes of ¨ Ž b1 . and ¨ Ž b 2 . is of order p 2 and one of the following three conditions is fulfilled: Ža.

Kˆ3rKˆ is a Galois extension of degree p 2 ;

Žb. char Kˆ s p and Kˆ3rKˆ is a purely inseparable simple extension of degree p 2 ;

ˆ and K 2 is an immediate extension of K. Žc. char Kˆ s p, Kˆ1 / K, Suppose further that K contains a primitive pth root of unity or char K s p. It is well known Žcf. wL 1 , Chap. VIII, Theorems 10 and 11x. that then there are elements a i g K * and j i g K i , such that K i s K Ž j i . and j i p y c. j i y a i s 0: i s 1, 2, where c s 0 in case of char K / p and c s 1, otherwise. Lemma 2.3, formula Ž1.3., and the noted properties of D9 imply that D 1 mK D 2 is a division algebra in each of the following special cases: Case 1. If the co-classes of ¨ Ž a1 ., ¨ Ž b1 ., ¨ Ž a2 ., and ¨ Ž b 2 . generate a subgroup of ¨ Ž K .rp.¨ Ž K . of order p 4 .

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Case 2. If the subgroup of ¨ Ž K .rp.¨ Ž K . generated by the co-classes of ¨ Ž a1 ., ¨ Ž b1 ., and ¨ Ž b 2 . is of order p 3 and one of the following two statements is true:

p 2.

Ži. ¨ Ž K 2 . s ¨ Ž K .; Žii. char Kˆ s p and the quotient group ¨ Ž K 3 .r¨ Ž K . is cyclic of order

Case 3. If char Kˆ s p, ¨ Ž b1 . s ¨ Ž b 2 . s 0, and if the extension K 3rK and the residue classes ˆ b1 , ˆ b 2 g Kˆ satisfy the following conditions: Ži. The quotient group ¨ Ž K 3 .r¨ Ž K . is cyclic of order p 2 or Kˆ3rKˆ is a purely inseparable simple extension of degree p 2 ; Žii. The degree of the field extension Kˆ3p Ž ˆ b1 , ˆ b 2 .rKˆ3p is equal to p 2 . Case 4. If char Kˆ s p, ¨ Ž b1 . f p.¨ Ž K 3 ., ¨ Ž b 2 . s 0, the residue class ˆb 2 g Kˆ is out of Kˆ3p and the extension K 3rK satisfies condition Ži. of Case 3. The four cases are considered without much difficulty because of the ˆ of Ž D9, ¨ 9. is a field. Moreover, fact that the residue division ring D9 ˆ ˆ ˆ D9 s K in Cases 1 and 2Žii., and D9rKˆ is a purely inseparable extension ˆ has no K-automorphism ˆ in Cases 3 and 4. Hence, D9 other than the identity except, possibly, in Case 2Ži.. The following lemma contains sufficient conditions for K to admit field extension K 1 , K 2 , and K 3 as in Case 3Ži.. LEMMA 2.4. Assume that Ž K, ¨ . is a Henselian ¨ alued field with a residue field Kˆ of characteristic p ) 0 and that K contains a primiti¨ e pth root of unity unless char K s p. Assume also that K has some of the following properties: Ži. char K s 0 and ¨ Ž p . f p.¨ Ž K .; Žii. char K s 0, ¨ Ž p . g p.¨ Ž K . and K contains an element p , such that ¨ Ž p . ) ¨ Žp . ) 0 and ¨ Žp . f p.¨ Ž K .; Žiii. Kˆ is not a perfect field and either char K s p or char K s 0 and ¨ Ž p . g p.¨ Ž K .; Živ. char K s p, ¨ Ž K . / p.¨ Ž K . and Kˆ has more than p elements. Then the algebraic closure K of K contains as subfields some cyclic extensions K 1 and K 2 of K of degree p, such that the compositum K 3 s K 1. K 2 satisfies condition Ži. of Case 3. Proof. The fields K 1 and K 2 can be obtained by adjoining to K some roots j 1 and j 2 in K of the equations x p y c. x y a i s 0: i s 1, 2, for certain elements a1 , a2 g K, c being equal to 0 or 1 depending on whether

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char K s 0 or p, respectively. The elements a1 and a2 can be determined as follows: ŽA. If char K s 0, then a2 s a1 q 1, where Ža. a1 s p in case of ¨ Ž p . f p.¨ Ž K .; Žb. a1 s p , if p and p are as in Lemma 2.4Žii.; Žc. ¨ Ž a1 . s 0 and the residue class a ˆ1 is not a pth power in ˆ ˆ K, provided that K is not a perfect field and ¨ Ž p . g p.¨ Ž K .. ŽB. If char K s p, then a2 s l p.a1 , for some l g K with ¨ Ž l. s 0 ˆ out of the prime subfield of K, ˆ and a1 satisfying one and a residue class l of the following two conditions: Ža. ¨ Ž a1 . - 0 and ¨ Ž a1 . f p.¨ Ž K ., if ¨ Ž K . / p.¨ Ž K .; 2 Žb. if the field Kˆ is not perfect, then a1 s byp .aX1 , for some elements b, aX1 g K *, such that ¨ Ž aX1 . s 0 - ¨ Ž b . and the residue class a ˆX1 ˆ is not a pth power in K. Denote by j 3 the element j 1 q 1 y j 2 when char K s 0 and put j 3 s l. j 1 y j 2 in case that char K s p. Observe that if a1 and a2 are fixed as required by ŽA. or ŽB., then the equations x p y c. x y a i s 0: i s 1, 2, have no root in K. In view of the elementary properties of the valuation ¨ , this is easily verified by straightforward calculations. Furthermore, it follows from Kummer’s theory and its analogue based on the Artin]Schreier theorem Žcf. wL 1 , Chap. VIII, Theorems 11, 13, 14, and 15; CF, Chap. III, Lemma 2.3x. that the fields K 1 s K Ž j 1 . and K 2 s K Ž j 2 . are distinct cyclic extensions of K of degree p; hence, their compositum K 3 is a noncyclic abelian extension of degree p 2 . Let ¨ 3 be the valuation of K 3 extending ¨ , and let j be the norm of j 3 from K 3 over K 1. A standard computation shows that j 3 s Ž j 1 q 1. p y j 2p if char K s 0, and j 3 s Ž l p y l.. j 1 if char K s p. This implies that if a1 and a2 are determined as in ŽA.Ža., ŽA.Žb., or ŽB.Ža., then ¨ 3 Ž j 3 . f ¨ Ž K .. Therefore, the equality ¨ 3 Ž j 3p . s ¨ 3 Ž j 3 . and formula Ž1.3. indicate that under any of these conditions, the quotient group ¨ Ž K 3 .r¨ Ž K . is cyclic of order p 2 and is generated by the co-class of ¨ 3 Ž j 3 .. Denote by j 3X the element b. j 3 if b, a1 and a2 are determined as in ŽB.Žb., and put j 3X s b . j 3 , for some element b g K such that ¨ Ž b p . s ¨ Ž py1 ., if a1 and a2 satisfy the condition ŽA.Žc.. Arguing as above, we obtain in each of these cases that ¨ 3 Ž j 3X . s 0, the residue class jˆ3X is a primitive element for ˆ and the field extension Kˆ3rKˆ is purely inseparable of degree Kˆ3 over K, p 2 . Lemma 2.4 is proved. Suppose that Ž K, ¨ . is a Henselian valued stable field. It follows from Lemma 2.4 and the discussion preceding it that K, ¨ , t Ž p ., and Kˆ satisfy the conditions of Theorem 2.1Ži.. Another result of this discussion is that if

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t Ž p . G 2, then the maximal p-extension KˆŽ p . of Kˆ contains as a subfield at most one cyclic extension of Kˆ of degree p. In view of Galois theory and wSe, Chap. I, Proposition 25x, this means that GŽ KˆŽ p .rKˆ. is a pro-p-group of rank F 1. Applying now wCh 4 , Remark 3.5Žii. and Lemma ˆ then GŽ KˆŽ p .rKˆ. ( Z p unless 3.8x, one concludes that if KˆŽ p . / K, p s 2 and Kˆ is formally real. Thus Theorem 2.1Žiii. reduces to a consequence of Theorem 2.1Živ.. We turn to the proof of Theorem 2.1Živ.. Let Kˆ be a formally real field and ¨ Ž K . / 2.¨ Ž K .. It is clear from Proposition 1.1 and wCh 4 , Lemma 3.8x that Kˆ is Pythagorean and KˆŽ2. s KˆŽ'y1 .. Taking also into consideration that K is formally real, one obtains that the equation x 2 q y 2 s y1 ˆ In view of wP, Sect. 15.1, Lemmax, this has no solution over K and K. indicates that Ay1Žy1, y1; K . is an inertial central division algebra over K. Note also that if t Ž2. G 2, and a and b are elements of K, such that the subgroup of ¨ Ž K .r2.¨ Ž K . generated by the co-classes of ¨ Ž a. and ¨ Ž b . is of order 4, then Ay1Ž a, b; K . is a totally ramified central division K-algebra. It is now easy to see from Ž1.3. and Lemma 2.3 that Ay1Žy1,y1; K . mK Ay1Ž a, b; K . is a division algebra, in contradiction with the stability of its centre K. As ¨ Ž K . / 2.¨ Ž K ., the obtained contradiction shows that t Ž2. s 1, which completes the proof of Theorem 2.1Živ.. It remains for us to prove Theorem 2.1Žii.. Assume that K contains a primitive pth root of unity and t Ž p . s 3. Lemma 2.4 and Case 2Ži. of the discussion preceding it, combined with formula Ž1.3., imply that then K and Kˆ satisfy the following conditions: Ž2.1. ¨ Ž K 9. / ¨ Ž K ., for every cyclic field extension K 9rK of degree p; in particular, Kˆ has no cyclic extensions of degree p. Moreover, if char Kˆ s p, then Kˆ is a perfect field and t.¨ Ž p . - ¨ Žp ., for all integers t and every p g K with ¨ Žp . ) 0 and ¨ Žp . f p.¨ Ž K .. It is clear from Ž2.1. as well as from the last assertion of Theorem 2.1Ži. ˆ Let c1 , c2 , and c3 be elements of K, such that the that KˆŽ p . s K. co-classes of ¨ Ž c1 ., ¨ Ž c 2 ., and ¨ Ž c 3 . generate the quotient group ¨ Ž K .rp.¨ Ž K .. As K contains a primitive pth root of unity, its maximal p-extension K Ž p . contains p n th roots of c1 , c 2 , and c 3 , for all n g N. Denote by K 0 Ž p . the extension of K generated by the set of these roots. We prove the remaining part of Theorem 2.1Žii. by establishing successively the following properties of K and K 0 Ž p .: Ž2.2. K contains a primitive p n th root of unity for every n g N, K 0 Ž p .rK is an abelian extension with a Galois group isomorphic to Z p3 , and K 0 Ž p . s K Ž p ..

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ˆ Since KˆŽ p . s K, ˆ one obtains in this case Suppose first that p / char K. from the Henselian property of Ž K, ¨ . and the availability of a primitive pth root of unity in K that Kˆ and K contain primitive p n th roots of unity, for every n g N. Hence, K 0 Ž p .rK is an abelian extension. More precisely, it follows from Kummer’s theory and the assumption on c1 , c 2 , and c 3 that the Galois group GŽ K 0 Ž p .rK . has a divisible character group and that a finite group G can be realized as a quotient group of GŽ K 0 Ž p .rK . by an appropriate open subgroup UG if and only if G is an abelian p-group of rank F 3. As the class of these abelian groups is closed under the formation of subgroups and quotient groups, it is easy to see from Galois theory that the considered properties of GŽ K 0 Ž p .rK . are preserved by GŽ K 0 Ž p .rK 9., for every finite extension K 9 of K in K 0 Ž p .. In view of wF, Theorem 23.1x and Pontrjagin’s duality wPo, Chap. 6; F, Theorem 23.1x, this indicates that the character group of GŽ K 0 Ž p .rK 9. can be presented as a direct sum of three isomorphic copies of the quasi-cyclic group ZŽ p` . and that there exists a topological group isomorphism GŽ K 0 Ž p .rK 9. ( Z p3. Taking once again into account that KˆŽ p . s Kˆ and observing that the residue field Kˆ9 of K 9 with respect to the valuation ¨ 9 extending ¨ is a ˆ one obtains that Kˆ9 s K. ˆ Since K contains a primitive p-extension of K, ˆ this implies that Kˆ9 s Kˆ9 p. Hence, by pth root of unity and p / char K, Ž1.2. and the Henselian property of ¨ 9, the quotient groups K 9*rK 9* p and ¨ 9Ž K 9.rp.¨ 9Ž K 9. are isomorphic. By Remark 2.2, these abelian p-groups are of rank t Ž p . s 3, so one sees from Kummer’s theory and the established properties of GŽ K 0 Ž p .rK 9. that every extension of K 9 in K Ž p . of degree p is a subfield of K 0 Ž p .. It is now easily proved by induction on the degree over K that every finite extension of K in K Ž p . is contained in K 0 Ž p .. This shows that K 0 Ž p . s K Ž p . and completes the proof of Ž2.2. in ˆ the case of p / char K. Assume now that char Kˆ s p. It suffices to show that there is a Henselian valuation of K with a residue field of zero characteristic and a value group satisfying the condition of Theorem 2.1Žii. with respect to p. Since the set I s  m g K: ¨ Ž m . ) t.¨ Ž p ., for all integers t 4 is a prime ideal of the valuation ring OK of Ž K, ¨ ., one can attach to it a valuation w of K in the same way as in wR, Sect. 1, ŽA7.x. The value group w Ž K . is the quotient group of ¨ Ž K . by the subgroup  h, yh g ¨ Ž K .: h s ¨ Ž b h ., for some element b h g Ž OK R I .4 . The ordering F1 on w Ž K . is canonically induced by the ordering F on ¨ Ž K . so that v Ž g 1 . F1 v Ž g 2 . whenever g 1 , g 2 g ¨ Ž K . and g 1 F g 2 , where v is the natural group homomorphism of ¨ Ž K . on w Ž K .. By definition, the valuation w acts on the elements of K * as the composition v ( ¨ , which indicates that it induces the trivial valuation on the prime subfield of K, or what amounts to the same, that the residue field of Ž K, w . is of characteristic zero. By Corollary 3 of wR,

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Sect. 2x, Ž K, w . is a Henselian valued field; also, it follows from Ž2.1. that the quotient group w Ž K .rp.w Ž K . is of rank 3, and that it is generated by the co-classes of w Ž c i .: i s 1, 2, 3. As the established properties of K and w imply Ž2.2., Theorem 2.1 is proved. COROLLARY 2.5. With the assumptions being as in Theorem 2.1Žii., then K Ž p .rK is a totally ramified extension. Proof. Let c1 , c 2 , and c 3 be elements of K determined as in the proof of Theorem 2.1Žiii.. It follows from Ž2.2. that the field K Ž p . can be X X presented as a union D`ms 1 K m , where K m is an extension of K generated by the p m th roots of c1 , c2 , and c 3 in K Ž p ., for every m g N. It becomes X clear from Ž1.3. that K m rK are totally ramified extensions, for all m g N. This proves Corollary 2.5, since every finite extension K 9 of K in K Ž p . is X X Ždepending on K 9. and e Ž K m a subfield of some K m rK . s X Ž . Ž . e K m rK 9 .e K 9rK . COROLLARY 2.6. Let Ž K, ¨ . be a Henselian ¨ alued stable closed field and ˆ and each Sylow pro-p-subgroup Gˆp of the p a prime number. Then t Ž p ., K, absolute Galois group GKˆ satisfy the following conditions:

ˆp / 14, then the open subgroups of Gˆp are Ži. If t Ž p . s 1 and G p-groups of Demushkin type, unless p s 2 and Kˆ is formally real; Žii. t Ž p . F 3 and the equality t Ž p . s 3 is possible only in the case of ˆp s 14. Moreo¨ er, if t Ž p . s 2 and Gˆp / 14, then Gˆp is char K / p and G isomorphic to Z p ; Žiii. Kˆ is an almost perfect field whene¨ er char Kˆ s p and t Ž p . G 1; furthermore, if char Kˆ s p and t Ž p . G 2, then Kˆ is perfect. Proof. Ži. and Žii. Let K p be an extension of K in K s e p corresponding by Galois theory to some Sylow pro-p-subgroup of GK . Then K p contains a primitive pth root of unity unless char K s p, and by wCh 4 , Lemma 4.3x, K p is a stable closed field. It follows from the general properties of inertial extensions of K in K s e p and of the Sylow subgroups ˆp is isomorphic of profinite groups Žcf. wJW, p. 135; Se, Chap. I, 1.4x. that G to the absolute Galois group of the residue field Kˆp of the valuation ¨ p of K p extending ¨ . Since the degrees of the finite extensions of K in K p are not divisible by p, Remark 2.2 indicates that the p-group ¨ p Ž K p .rp.¨ p Ž K p . is of rank t Ž p .. It is now clear from Proposition 1.1 and the proof of Corollary 2.6 in wCh 4 x that Kˆp is a quasi-local field, provided that t Ž p . ) 0. These observations show that Corollary 2.6Ži. ] Žii. can be obtained as a consequence of Proposition 1.2Ži. and Theorem 2.1, applied to Kˆp and Ž K p , ¨ p ., respectively. Žiii. If char K s p, our conclusion is contained in Theorem 2.1Ži. and Žii., so it suffices to consider the special case of char K s 0 and char Kˆ s p.

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ˆL ˆ p x s w K: ˆ Kˆ p x, for every finite extension Since w L: tained from Ž1.3. and Remark 2.2 that one can also that K contains a primitive pth root of unity. In this reduces to a special case of Theorem 2.1Ži. and Žii..

ˆ ˆ it is easily obLrK, assume for the proof way, Corollary 2.6Žiii.

COROLLARY 2.7. Let Ž K, ¨ . be a Henselian ¨ alued stable closed field with ˆ be a central di¨ ision algebra o¨ er K. ˆ a totally indi¨ isible ¨ alue group, and let D ˆ of Kˆ of degree di¨ iding indŽ Dˆ. embeds in Dˆ as a Then e¨ ery extension L ˆ K-subalgebra. Proof. By Corollary 2.6 of wCh 4 x, and Theorem 2.1Ži., Kˆ is a quasi-local almost perfect field. This, of course, also applies to the separable closure ˆ 0 of Kˆ in L. ˆ Note further that LrL ˆ ˆ 0 is a purely inseparable extension L whenever char Kˆ / 0 Žcf. wL 1 , Chap. VII, Proposition 10x.. These observations show that our assertion can be deduced from wCh 4 , Corollary 4.7 and ˆ ˆ mKˆ L ˆ ( Ž Dˆ mKˆ L ˆ 0 . mLˆ 0 L. ˆ Lemma 4.4x , and the L-isomorphism D 3. A SUFFICIENT CONDITION FOR STABILITY OF HENSELIAN VALUED FIELDS The main result of this section shows that a Henselian valued field Ž K, ¨ . is stable whenever char Kˆ s 0 and the properties of Kˆ and ¨ Ž K . are admissible by Proposition 1.1 and Theorem 2.1. On the other hand, it is not difficult to see that Proposition 1.1 and Theorem 2.1 do not provide generally valid sufficient conditions for stability of K. For example, statements wTY, Theorem 4.1x Žwith its proof., Ž1.3., and wP, Sect. 13.4, Corollaryx imply, for each prime q, the existence of a Henselian valued field Ž L q , wq . of characteristic q with an algebraically closed residue field and a divisible value group, admitting Žimmediate. central division L q-algebras of exponent q and index q n, for every n g N. This gives a good idea of the possible difficulties in analyzing the relations between Schur indices and exponents of central division K-algebras of q-primary dimensions when char Kˆ s q. In this section we prove that if the properties of Kˆ and ¨ Ž K . are admissible by Proposition 1.1 and Theorem 2.1, then only these relations can be an obstruction to the stability of K. We also show that this obstruction vanishes in several interesting special cases. THEOREM 3.1. Assume that Ž K, ¨ . is a Henselian ¨ alued field with a residue field Kˆ and in¨ ariants t Ž p . of the ¨ alue group ¨ Ž K . satisfying the stability conditions established by Proposition 1.1 and Theorem 2.1, for e¨ ery prime number p. Then: Ža. K is a stable field if and only if either char Kˆ s 0 or char Kˆ s q ) 0 and expŽ D q . s indŽ D q ., for e¨ ery central di¨ ision K-algebra D q of q-power dimension;

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Žb.

357

K is stable in each of the following special cases: Ži.

ˆ K is an almost perfect field and char K s char K;

Žii. Kˆ is a perfect field of characteristic q ) 0, t Ž q . F 1, and all central di¨ ision K-algebras are defectless; Žiii. char Kˆ s q / char K, t Ž q . s 3, and K contains a primiti¨ e qth root of unity. Proof. Theorem 3.1Ža. is equivalent to the statement that expŽ D . s indŽ D . s p k , for every central division K-algebra D of dimension not ˆ It suffices to prove this statement in the special case of divisible by char K. k Ž ind D s p , where p is a prime number not equal to char Kˆ and k g N. Our proof is based on the following well-known result Žimplied by wJW, Decomposition Lemmas 5.14 and 6.2, Theorem 5.15, and Corollary 1.5x.: Ž3.1. D is similar to the K-algebra S mK V mK T, for some central division K-algebras S, V, and T of p-power dimensions, such that S is inertial and T is totally ramified over K, V possesses a maximal subfield V1 inertial over K, and another maximal subfield V2 totally ramified over K. In particular, Vˆ1 s Vˆ and ¨ Ž V2 . s ¨ Ž V .. Moreover, if D is the underlyˆ ing division K-algebra of S mK V, then ¨ Ž V2 . s ¨ Ž D . and Vˆ1 is K-isomorˆ. of D. ˆ phic to the centre ZŽ D We first prove that V is a cyclic K-algebra, indŽ V . s expŽ V . and indŽ D . s expŽ D .. It follows from wJW, Theorem 2.8Žb.x and the properties ˆ since Kˆ is of S that the residue division ring Sˆ is a central algebra over K; assumed to be stable, the quoted theorem also shows that indŽ S . s indŽ Sˆ. s expŽ Sˆ. s expŽ S .. As Ž1.3. implies that T s V s K and D ( D ( S in the special case of ¨ Ž K . s p.¨ Ž K ., one can assume further that ¨ Ž K . / p.¨ Ž K .. In view of the Henselian property of Ž K, ¨ ., formula Ž1.3., and ˆ.rKˆ and V1rK are abelian p-extensions with wJW, Proposition 1.7x, ZŽ D Galois groups isomorphic to the quotient group ¨ Ž D .r¨ Ž K . s ¨ Ž V .r¨ Ž K .. Clearly, ¨ Ž D .r¨ Ž K . is a cyclic group, if t Ž p . F 1; so is the Galois group ˆ.rKˆ. in case of t Ž p . G 2, by stability condition Žc. of Theorem 2.1. GŽ Z Ž D ˆ.rKˆ and V1rK are cyclic field extensions of degree indŽ V . s Hence, ZŽ D k1 p , so it follows from Ž3.1. and wP, Sect. 15.1, Proposition a; JW, Proposition 1.7x, that V can be presented as a cyclic K-algebra Ž V1 , s 1 , n ., where s 1 is a K-automorphism of the field V1 of order p k 1 , and n is some element of K *, for which ¨ Ž n . f p.¨ Ž K .. As the group ¨ Ž V .r¨ Ž K . is cyclic of order p k 1 , one obtains from wP, Sect. 14.4, Proposition bŽii.; PY, Ž3.19.x, that expŽ V . s p k 1 s indŽ V .. Put indŽ S . s indŽ Sˆ. s p k 2 , k s ˆ. possesses a min k 1 , k 2 4 , and k9 s max k 1 , k 2 4 . By Galois theory, ZŽ D k subfield Kˆ9 which is a cyclic extension of Kˆ of degree p ; the field Kˆ9 can

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ˆ be embedded as a K-subalgebra in Sˆ since ¨ Ž K . / p.¨ Ž K . and central ˆ division K-algebras of p-primary dimensions satisfy the stability condition ˆ. and established by Proposition 1.1. As the algebras Ž Sˆ mKˆ Kˆ9. mKˆ9 ZŽ D ˆ. are isomorphic over ZŽ Dˆ. Žcf. wP, Sect. 9.4, Corollary ax., one Sˆ mKˆ ZŽ D sees from Wedderburn’s structure theorem and wCh 1 , Lemma 3.5x that the ˆ. is of index p k 2yk over its centre underlying division algebra of Sˆ mKˆ ZŽ D ˆ.. Taking also into consideration that ¨ Ž D .r¨ Ž K . is a cyclic group of ZŽ D order p k 1 , one obtains from wJW, Theorem 5.15x, that indŽ D . s expŽ D . s p k 9. This proves that indŽ D . s expŽ D . in the case of T s K, since D is then K-isomorphic to D. Assuming further that T / K and expŽT . s p t , we show that K, S, T, and V have the following properties: Ž3.2. Ži. K contains a primitive root of unity e of degree p t , 2 F t Ž p . F 3 and either KˆŽ p . s Kˆ or t Ž p . s 2 and KˆŽ p .rKˆ is a Z p-extension; Žii. S s K, T is a symbol K-algebra, and indŽT . s p t. Moreover, if T s Ae Ž l, m ; K . for some l, m g K *, then the group ¨ ŽT .r¨ Ž K . is generated by the co-classes of pyt .¨ Ž l. and pyt .¨ Ž m .; Žiii. V is isomorphic to the cyclic K-algebra Ž V1 , s 1 , n 1 ., for some element n 1 g K which is a p t th power in T but is not a pth power in K. It follows from wJW, Theorem 1.10x and the assumptions on T that Ž t p . G 2 and K contains a primitive p t th root of unity e . Since K satisfies the stability conditions established by Theorem 2.1, this proves Ž3.2.Ži.. Applying Ž3.2.Ži. and Galois cohomology Žcf. wSe, Chap. I, Sect. 4x., one obtains that H 2 Ž GŽ KˆŽ p .rKˆ., Fp . s  04 . By the Merkurjev]Suslin theorem Žcf. wMS, Ž11.5. and Sect. 9x or wS, Sects. 18, 19, and 21x., this means that Br Ž Kˆ. p s  04 ; hence, Sˆ s Kˆ and S s K. As t Ž p . F 3, it is not difficult to see from Ž1.3. that ¨ ŽT .r¨ Ž K . is an abelian p-group of rank F 3. More precisely, Theorem 1.10 of wJWx, this observation, and the fact that T is totally ramified over K indicate that ¨ ŽT .r¨ Ž K . is isomorphic to the direct sum of two cyclic groups of order indŽT .. It is therefore clear from Proposition 3.14 of wPYx, with its proof, that T has the properties required by Ž3.2.Žii.. Statement Ž3.2.Žiii. is now obvious for V s K, so we prove it assuming that V / K and T s Ae Ž l, m ; K ., for some l, m g K *. We have already seen that V can be identified with the cyclic K-algebra Ž V1 , s 1 , n ., for some n g K * of value ¨ Ž n . f p.¨ Ž K .. The assumption that V / K ˆ In view of Ž3.1. and Ž3.2.Ži. ] Žii., this shows that V1 / K and KˆŽ p . / K. means that t Ž p . s 2 and the co-classes of ¨ Ž l. and ¨ Ž m . generate the group ¨ Ž K .rp.¨ Ž K .. Therefore, n can be presented as a product n s k li. m j.n 9.n p 1 , for suitably chosen elements n g K * and n 9 g K * with ¨ Ž n 9. s 0, and some integers i, j G 0 with a greatest common divisor relatively prime to p. It is clear from the definition of V that it is

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K-isomorphic to Ž V1 , s 1 , li. m j.n 9., i.e., n can be taken so that n s 1. As V1rK is an inertial cyclic p-extension and Br Ž Kˆ. p s  04 , one obtains from wJW, Theorem 5.6x that the K-algebra Ž V1 , s 1 , u. is similar to K, whenever u g K and ¨ Ž u. s 0. By the lemma in wP, Sect. 15.1x, this is equivalent to the assertion that the norm group N Ž V1rK . contains all elements of K of value zero. This, applied to the elements n 9 and yn 9, also implies that V is isomorphic to the K-algebras Ž V1 , s 1 , li. m j . and Ž V1 , s 1 , yli. m j .. At the same time, it follows from the definition of T that at least one of the elements li. m j and yli. m j is a p t th power in T. Note also that li. m j and yli. m j are not pth powers in K, since ¨ Ž li, m j . s ¨ Žyli. m j . is not an element of p.¨ Ž K .. Thus Ž3.2. is proved. We are now ready to show that indŽ D . s expŽ D . in the case of T / K. Our argument is based on the fact that S s K, i.e., D is similar to V mK T. Let K be an algebraic closure of K and n 1 an element of K fixed as in Ž3.2.Žiii.. Put m s min t, k 1 4 , m9 s max t, k 1 4 , and denote by K 1 the extension of K in K generated by some p m th root of n 1. Clearly, w K 1: K x s p m and K 1 is isomorphic to some K-subalgebras of V and T. This, combined with wCh 1 , Lemma 3.5x and the general properties of tensor products Žcf. wP, Sect. 9.3, Corollary b, and Sect. 9.4, Corollary ax., proves the existence of the following sequences of isomorphisms of K 1-algebras,

Ž V mK T . mK K 1 ( Ž Ma Ž D . . mK K 1 ( Ma Ž D mK K 1 . ( Ma Ž Mb Ž D 1 . . ( Mab Ž D 1 . and

Ž V mK T . mK K 1 ( Ž V mK K 1 . mK 1 Ž T mK K 1 . ( M p m Ž V1X . mK 1 M p m Ž T1 . ( M p 2 m Ž V1X mK 1 T1 . ( M p 2 m Ž Ma9 Ž DX1 . . ( M p 2 m .a9 Ž DX1 . , for some central division K 1-algebras D 1 , DX1 , V1X , and T1 , and some positive integers a, b, and a9, such that a < p 2 m and b < p m . By Wedderburn’s structure theorem, D 1 and DX1 are isomorphic over K 1 and a.b s p 2 m .a9. Hence, p m < a and indŽ D .< p m9. Note that if t / k 1 , then expŽ D . s p m9 since expŽ V . s p k 1 and expŽT . s p t ; in view of the above and wP, Sect. 14.4, Proposition bŽii.x, this shows that indŽ D . s p m9. Assume now that t s k 1 s m. In this case, V1 is a Kummer extension of K generated by a p t th root of some element r g K. As V1 is inertial over K, r can be chosen so that ¨ Ž r . s 0. Since the K-algebras Ž V1 , s 1 , n 1 . and Ž V1 , s 1n, n 1n . are isomorphic, for every integer n not divisible by p Žcf. wP, Sect. 15.1, Corollary ax., one can assume without loss of generality that s 1Ž r . s ey1 . r . In other words, V can be identified with the symbol K-algebras Aey1 Ž r , n 1; K . and Ae Ž n 1 , ry1 ; K .. Moreover, statements

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Ž3.2.Žiii. and wP, Sect. 15.1, Propositions a, b, and Lemmax imply that T can be identified as a K-algebra with Ae Ž n 1 , m 1; K ., and D with the underlying division algebra of Ae Ž n 1 , ry1 . m 1; K ., for some m 1 g K *. The valuation of the field K 1 extending ¨ Žand also denoted by ¨ . is unique, so we have ¨ Ž u . s ¨ Ž s Ž u .., for every K-automorphism s of K 1 and all u g K 1; it follows from Ž1.3. and Ž3.2.Žiii. that the group ¨ Ž K 1 .r¨ Ž K . is cyclic of order p t and is generated by the co-class of ¨ Ž n 1 ., n 1 being a p t th root of n 1 in K 1. Therefore, each element g g N Ž K 1rK . can be presented as a t product g s n 1j.g 1p .g 9, for some number j g  0, 1, . . . , p t y 14 and some elements g 1 g K * and g 9 g K * with ¨ Žg 9. s 0. Applying now the second ty 1 assertion of Ž3.2.Žii. to n 1 and m 1 , one obtains that m 1p f N Ž K 1rK . and ty 1 Ž ry1. m 1 . p f N Ž K 1rK .. Hence, by wP, Sect. 15.1, Corollary dx, Ae Ž n 1 , ry1 . m 1; K . is a division K-algebra Žisomorphic to D . of exponent and index p t. This completes the proof of Theorem 3.1Ža.. The proof of Theorem 3.1Žb. is much easier. Theorem 3.1Ža. shows that indŽ D . s expŽ D ., provided that D is a central division K-algebra and ˆ This, combined with wA, Chap. VII, indŽ D . is not divisible by char K. Theorem 22; Ch 4 , Lemma 4.4x, proves Theorem 3.1Žb.Ži.. It follows from the proof of Ž2.2. that if char K s 0, char Kˆ s q ) 0 and t.¨ Ž q . - ¨ Žp . for all integers t and every p g K *, such that ¨ Žp . ) 0 and ¨ Žp . f q.¨ Ž K ., then ¨ induces on K a Henselian valuation w with a residue field k of zero characteristic and a value group w Ž K ., for which w Ž K .rq.w Ž K . is a q-group of rank t Ž q .. Using the second part of the proof of Ž2.2., one also obtains that if K and ¨ Ž K . satisfy the stability conditions of Theorem 2.1Žii. with respect to q, then the same applies to K, k, and w Ž K .. Thus Theorem 3.1Žb.Žiii. reduces to a special case of Theorem 3.1Ža.. Suppose now that Kˆ is a perfect field, char Kˆ s q ) 0, t Ž q . F 1, and D q is a defectless central division algebra over K of q-power dimension. The condition t Ž q . F 1 and Ž1.3. imply that the quotient group ¨ Ž D q .r¨ Ž K . is cyclic and D q possesses a subfield Tq containing K and such that ¨ ŽTq . s ¨ Ž D q .; since w Tq : K x
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COROLLARY 3.2. Let K be a perfect field with a Henselian ¨ aluation ¨ , ˆ Then K is such that the ¨ alue group ¨ Ž K . is di¨ isible and char K s char K. ˆ stable if and only if so is K. Proof. Corollary 3.2 follows from wJW, Theorem 2.8Žb.x, since the degrees of central division K-algebras are not divisible by char Kˆ and therefore, Decomposition Lemmas 6.8 and 5.14 of wJWx imply that these algebras are inertial over K. COROLLARY 3.3. Assume that Ž K, ¨ . is a Henselian ¨ alued field satisfying the following conditions: Ži. Kˆ is perfect and e¨ ery central di¨ ision K-algebra is defectless; Žii. the quotient groups ¨ Ž K .rp.¨ Ž K . are of order p, for all prime numbers p.

ˆ Then K is a stable field if and only if Kˆ is stable and each cyclic extension L ˆ ˆ ˆ of K is embeddable as a K-subalgebra in e¨ ery central di¨ ision K-algebra of ˆ Kˆx. Schur index di¨ isible by w L: Proof. This is an immediate consequence of Theorem 3.1 and wCh 4 , Corollary 2.5x. PROPOSITION 3.4. Let Ž K, ¨ . be a Henselian ¨ alued stable field satisfying some of the following two conditions:

ˆ Ži. K is an almost perfect field and char K s char K; Žii. Kˆ is a perfect field, char Kˆ s q ) 0, t Ž q . F 1, and e¨ ery finite separable extension of K is defectless. Then e¨ ery totally ramified algebraic extension of K is a stable field. To prove Proposition 3.4Žii., we need the following lemma. LEMMA 3.5. Let Ž K, ¨ . be a Henselian ¨ alued field with a residue field Kˆ of characteristic q ) 0 and a ¨ alue group for which t Ž q . F 1. Then the following conditions are equi¨ alent: Ži. Žii. centre.

E¨ ery finite separable extension of K is defectless; E¨ ery finite dimensional di¨ ision K-algebra is defectless o¨ er its

Proof. The implication Žii. ª Ži. follows from wTY, Proposition 3.1; M, Sect. 4, Theorem 2x, so we prove that Ži. ª Žii.. In view of the general structure of central simple algebras Žcf. wP, Sect. 14.4x., and of statements Ž1.3. and wJW, Corollary 1.5x, it suffices to establish the defectlessness of central division algebras of q-primary dimensions over an arbitrary finite extension L of K. Lemma 3.4 of wTYx and Proposition 14 of wL 1 , Chap.

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XIIx indicate that L satisfies condition Ži. of the lemma with respect to its valuation ¨ L extending ¨ ; hence, by Theorem 3.1 of wTYx, central division L-algebras of index q are defectless. Taking also into account that the group ¨ LŽ L.rq.¨ LŽ L. is of order q t Ž q. Žsee Remark 2.2. and arguing as in the proof of Theorem 3.1Žb.Žii., one obtains that each algebra of this kind contains as a maximal subfield an inertial extension of L. This, combined with Theorem 2 of wM, Sect. 4x, the general properties of tensor products, and the basic results about inertial extensions of L in L s e p Žcf. wP, Sect. 9.4, Corollary a; and JW, p. 135x., implies that every central division Lalgebra of exponent q has a splitting field inertial over L. Suppose now that D is a central division L-algebra of exponent q n, for some n g N, and denote by D the q ny 1 th tensor power of D over L. Since D is a central simple L-algebra of exponent q, D mL M is a central simple M-algebra of exponent dividing q ny 1 whenever M is a splitting field of D. Proceeding by induction on n, one obtains now without difficulty that D is split over L by some inertial extension of L. Therefore, by Lemma 5.1 of wJWx, D is defectless over L, so Lemma 3.5 is proved. Proof of Proposition 3.4. Let L be a totally ramified algebraic extension of K and ¨ Ž L. the value group of the valuation ¨ L on L extending ¨ . Arguing as in the proof of Lemma 4.3 of wCh 4 x, one obtains that it suffices to establish the stability of L on the additional assumption that LrK is a finite extension. In this case, by Remark 2.2, the p-groups ¨ Ž L.rp.¨ Ž L. and ¨ Ž K .rp.¨ Ž K . are isomorphic for each prime p. As K is stable and Kˆ is the residue field of Ž L, ¨ L ., this means that L and ¨ L satisfy the necessary conditions for stability given by Theorem 2.1. Note also that K is an almost perfect field if and only if L is one. These observations, combined with Theorem 3.1 and Lemma 3.5, prove our assertion.

4. CHARACTERIZATIONS AND EXAMPLES OF HENSELIAN VALUED STABLE FIELDS First we show that the spectrum of invariants t Ž p . of Henselian valued stable fields with a given quasi-local perfect residue field F0 is fully determined by the restrictions established by Theorem 2.1, when p runs through the set of all prime numbers not equal to char F0 . In view of Corollary 2.6, this gives a possibility of constructing a series of Henselian valued stable fields which are not stable closed Žsee also Remark 4.2Žii... PROPOSITION 4.1. Let F0 be a quasi-local perfect field of characteristic q G 0 and let  nŽ p .: p g P0 4 be a sequence of integers G 0, indexed by a set of prime numbers P0 containing q, if q / 0, and including the union P Ž F0 . s P1Ž F0 . j P2 Ž F0 ., where P1Ž F0 . is the set of all prime numbers p9, for which

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F0 contains a primiti¨ e p9th root of unity, and P2 Ž F0 . consists of these primes p0, for which the Galois group of the maximal p0-extension F0 Ž p0 . of F0 is not procyclic. Assume that the numbers nŽ p . satisfy the following conditions: Ži. nŽ p . F 1, for all p g P2 Ž F0 ., also for p s q in case of q / 0, and for p s 2 in case that F is formally real; Žii. nŽ p . F 3 y r Ž p ., for e¨ ery p g Ž P Ž F0 . R P2 Ž F0 .., r Ž p . being equal to 1, if F0 Ž p . / F0 , and to 0, otherwise. Then there is a Henselian ¨ alued stable field Ž K, ¨ . of characteristic q with a residue field F0 and a ¨ alue group ¨ Ž K . whose in¨ ariants t Ž p . are equal to nŽ p ., for all p g P0 , and to infinity, for all primes p f P0 . Proof. We show that a field K with the required properties can be found among the algebraic extensions of the union F s D`ns 1 Fn , where  Fn s Fny1ŽŽ z n ..: n g N 4 is an inductively defined tower of Laurent formal power series fields. Let Z` be an inversely lexicographically ordered countable direct sum of isomorphic copies of the additive group Z of integer numbers, and let ¨ n be the standard Z n-valued valuations of the fields Fn acting trivially on F0 . Since ¨ n is Henselian and is extended to Fnq 1 by ¨ nq1 , for each n g N, the sequence  ¨ n : n g N 4 defines in an obvious manner a Henselian valuation ¨ F of F with a residue field F0 and a value group isomorphic to Z` . We denote by ¨ the unique valuation extending ¨ F of each intermediate field of the extension FrF, F being an algebraic closure of F, and we associate with every p g P0 a set of elements S p s hjŽ p.Ž m. g F: m g N, jŽ p . g N, jŽ p . ) nŽ p .4 , such that hjŽ p.Ž1. p s x jŽ p. and hjŽ p.Ž m. p s hjŽ p.Ž m y 1., for every pair of integers jŽ p . ) nŽ p ., m G 2. It is easy to see that the extension K p of F generated by S p is totally ramified and that each finite extension of F in K p is of p-power degree, for every p g P0 . Moreover, the p-group ¨ Ž K p .rp.¨ Ž K p . is of rank nŽ p . while the ranks of the p9-groups ¨ Ž K p .rp9.¨ Ž K p . are infinite for all prime numbers p9 / p. Therefore, the compositum K of the fields K p , for all p g P0 , is also totally ramified over F, F0 is the residue field of Ž K, ¨ ., the ranks of ¨ Ž K .rp.¨ Ž K . are equal to nŽ p . when p runs over the elements of P0 , and to infinity for every prime p out of P0 . As the field F0 is algebraically closed in F, this indicates that it is algebraically closed in K as well; in particular, all roots of unity in K are elements of F0 . In view of the assumptions in Proposition 4.1 and the established properties of ¨ Ž K ., this implies that the triple Ž K, ¨ Ž K ., Kˆ s F0 . satisfies the conditions of Proposition 1.1 and Theorem 2.1 with respect to every prime number p, so it follows from Theorem 3.1 that K is a stable field. Proposition 4.1 is proved. Remark 4.2. Ži. With the assumptions and notations being as in Proposition 4.1, assume also that P0 is the set of all prime numbers and nŽ p . F n, for some n g N. It is clear from the proof of Proposition 4.1

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that the field K can be obtained as a totally ramified algebraic extension of the iterated formal power series field Fn in n variables over F. Žii. Let  nŽ p .: p g P 4 be a sequence of elements of N j  0, `4 , indexed by the set P of all prime numbers. Theorem 2.1 and Proposition 4.1 indicate that if F0 is a field of characteristic zero not containing a primitive pth root of unity, for any p g Ž P R  24., and such that GF 0 is a metabelian group of cohomological dimension one Žcf. wCh 2 , Example 1.3; Ch 3 , Sect. 2x., then there exists a Henselian valued stable field with a residue field F0 and invariants t Ž p . s nŽ p ., for all p g P, if and only if nŽ2. F 2. Similarly, if F0 is a finite field with q m elements and nŽ q . F 1, where q s char F0 , then we have nŽ p . s t Ž p .: p g P, for some Henselian valued stable field with a residue field F0 , if and only if nŽ p . F 2, for every p g P dividing q m y 1. In the rest of this section, we characterize the stable closed fields in some of the most interesting classes of Henselian valued fields. We consider almost exclusively nonreal fields, since the case of formally real fields with 2-indivisible value groups has been an object of a detailed research in Section 3 of wCh 2 x. PROPOSITION 4.3. Let Ž K, ¨ . be a Henselian ¨ alued field satisfying at least one of the conditions Ži. and Žii. of Proposition 3.4. Then K is stable closed if ˆ GKˆ, and ¨ Ž K . ha¨ e the following properties: and only if K, Žj. Kˆ is stable closed; Žjj. t Ž p . F 3, for e¨ ery prime number p; Žjjj. If t Ž p . G 1 and cd p Ž GKˆ . ) 0, for some prime p, then t Ž p . F 2 and each open subgroup P of a Sylow pro-p-subgroup of GKˆ is of Demushkin type; moreo¨ er, P is isomorphic to Z p in case t Ž p . s 2.

ˆ GKˆ, Proof. We show that K is a stable closed field, provided that K, and ¨ Ž K . have properties Žj., Žjj., and Žjjj.; the inverse implication follows from Proposition 1.1, Corollary 2.6, and the fact that Z p is continuously isomorphic to its open subgroups. It is not difficult to see from wCh 4 , Lemma 4.5x, formula Ž1.3., and the proof of Corollary 2.6 that one may consider only the special case in which GK is a pro-p-group, for some ˆ our assertion can be deduced from Theorem 3.1, prime p. If p s char K, ˆ Since GKˆ is a Lemma 3.5, and Remark 2.2. Suppose that p / char K. pro-p-group, we obtain from Žjjj. and wCh 4 , Lemma 3.11x, that Kˆ is a quasi-local field, provided that t Ž p . G 1. It is therefore clear from Ž1.3., ˆ GKˆ, and ¨ Ž K . that the properties Remark 2.2, and our assumptions on K, of the value groups and residue fields of valued finite extensions of Ž K, ¨ . are admissible by Proposition 1.1 and Theorem 2.1. This, combined with ˆ Theorem 3.1, implies that K is a stable closed field when p / char K, which completes our proof.

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COROLLARY 4.4. Let K s KˆŽŽ X 1 ..ŽŽ X 2 .. . . . ŽŽ X n .. be an iterated formal ˆ Then the Laurent power series field in n G 3 indeterminates o¨ er a field K. following statements are true: Ži. K is a stable field if and only if n s 3, char Kˆ s 0, and Kˆ does not admit proper abelian extensions; Žii. K is stable closed if and only if n s 3, char Kˆ s 0, and Kˆ is algebraically closed. Proof. We consider K with respect to its standard Z n-valued valuation. This valuation is Henselian with a residue field Kˆ and invariants t Ž p . s n, for every prime p. Hence, by Theorem 2.1Ži., applied to the case of p s 2, K is not a stable field unless n s 3 and char Kˆ s 0. Our assertions concerning the case of n s 3 follow from Theorem 2.1Žii., Proposition 4.3, and the elementary properties of cyclotomic extensions and Euler’s wfunction Žcf. wL 1 , Chap. VIII, Sect. 3; IR, Chap. 4, Sect. 1x.. COROLLARY 4.5. Let K s KˆŽŽ X ..ŽŽ Y .. be an iterated formal power series ˆ and let P Ž Kˆ. be the set of all prime field in two indeterminates o¨ er a field K, ˆ ˆ numbers for which K Ž p . / K. Then the following is true: Ži. K is a stable field if and only if Kˆ is perfect and stable, central ˆ di¨ ision K-algebras of index p are cyclic, and the Galois group GŽ KˆŽ p .rKˆ. is isomorphic to Z p , for e¨ ery p g P Ž Kˆ.; Žii. K is stable closed if and only if Kˆ is a perfect field with a metabelian absolute Galois group of cohomological dimension F 1. Proof. If char Kˆ s 0, our assertion can be proved by applying Theorem 3.1 and Proposition 4.3 to the standard Z 2-valued valuation of K. Similarly, the necessity of the conditions on Kˆ in statements Ži. and Žii. follows from Proposition 1.1, Theorem 2.1, and Corollary 2.6. Conversely, the sufficiency of the condition on Kˆ in Ži. can be obtained as a consequence of Theorem 3.1 and the following statement: Ž4.1. Let K be an iterated formal power series field in 2 indeterminates over a perfect field Kˆ of characteristic q ) 0, such that either KˆŽ q . s Kˆ or GŽ KˆŽ q .rKˆ. is isomorphic to Z q . Then indŽ D . s expŽ D ., for every central division K-algebra D of q-primary dimension. Let D be a central division algebra of exponent q n over a field K satisfying the conditions of Ž4.1.. For the case of n s 1, Tignol has established Žcf. wAJx. that D is a cyclic K-algebra of index q, so one may assume further that n G 2. Tignol’s result, applied to the underlying division algebra of the q ny 1 th tensor power of D over K, shows that there is an extension K 9 of K in K Ž q . of degree q, such that D mK K 9 is a central simple K 9-algebra of exponent q ny 1. It is not difficult to see from

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the general properties of complete discrete valued fields Žcf. wE, Ž5.7., Ž5.9., and Ž18.8.x. that K 1 is isomorphic to an iterated formal power series field in 2 indeterminates over an extension Kˆ1 of Kˆ of degree 1 or q. Since Kˆ9Ž q . s KˆŽ q . and GŽ KˆŽ q .rKˆ9. is isomorphic to Z q whenever Kˆ9 / Kˆ Žcf. wK, Chap. 9; Se, Chap. I, 4.5x., this enables one to prove inductively the existence of a subfield D9 of K Ž q . that is a splitting field of D, and an extension of K of degree q n. Statement Ž4.1. follows from the obtained result and the basic theory of simple algebras Žcf. wP, Sects. 13.4 and 14.4x., so Corollary 4.5Ži. is proved. Observing finally that the Brauer groups of the finite extensions of Kˆ are trivial in case Kˆ is perfect with cdŽ GKˆ . F 1 Žcf. wSe, Chap. II, Proposition 6Žb.x, and also, that every finite extension L of K is isomorphic to an iterated formal power series field in 2 indetermiˆ of K, ˆ one concludes that Corollary 4.5Žii. nates over a finite extension L can be deduced from Corollary 4.5Ži., Galois theory, and wCh 3 , Lemma 1.2x. This completes our proof. EXAMPLE. Let Q be an algebraic closure of the field Q of rational numbers, and let Q0 be the compositum of all normal finite extensions of Q in Q with solvable Galois groups. It is well known Žcf. wSe, Chap. II, 3.1x. that Q0 is not an algebraically closed field, and that it does not admit proper abelian extensions. Therefore, by Corollary 4.4, the power series field Q0 ŽŽ X ..ŽŽ Y ..ŽŽ Z .. is stable but not stable closed. The field Q0 is a normal extension of Q, so it follows from wCh 3 , Lemmas 1.2 and 3.5x that every profinite abelian group G of cohomological dimension F 1 is isomorphic to the Galois group GŽ Q0rQ G ., for some subfield Q G of Q0 . It is clear from Galois theory and our assumptions that G is isomorphic to the topological group product Ł p GŽ Q G Ž p .rQ G ., taken over the set of prime numbers. In particular, GŽ Q G Ž p .rQ G . is isomorphic to the Sylow pro-p-subgroup of G, for each prime p; in view of wCh 3 , Lemma 1.2x, this means that either Q G Ž p .rQ G is a Z p-extension or Q G Ž p . s Q G . Note also that each central division Q G-algebra D is cyclic with expŽ D . s indŽ D .; since D is a locally finite dimensional algebra over Q, this can be deduced from the stability of algebraic number fields, and the cyclicity of their central division algebras Žcf. Lemma 4.3 of wCh 4 x, with its proof.. These observations, combined with Corollary 4.5, indicate that the field Q G ŽŽ X ..ŽŽ Y .. is stable but not stable closed. COROLLARY 4.6. A Henselian discrete ¨ alued field Ž K, ¨ . with a perfect residue field Kˆ is stable closed if and only if Kˆ is quasi-local. Proof. Since finite separable extensions of K are defectless Žcf. wTY, Propositions 2.2 and 3.1x. and t Ž p . s 1, for every prime p, this can be deduced from wCh 2 , Proposition 3.1x, and Propositions 4.3Žjjj. and 1.2. Our

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conclusion also follows from Corollary 3.3 and wCh 4 , Corollary 2.6; TY, Proposition 2.2x. COROLLARY 4.7. Let P be the set of prime numbers and  nŽ p .: p g P 4 a sequence of nonnegati¨ e integer numbers. Then there exists a stable closed field K, such that the p-component of Br Ž K . is isomorphic to a direct sum of a di¨ isible group and a cyclic group of order p nŽ p., for each p g P. Corollary 4.7 shows that p nŽ p.. Br Ž K . p is the maximal divisible subgroup of Br Ž K . p , for every p g P; this gives an affirmative answer to Question 2 of wFSx. Proof. It follows from wCh 3 , Proposition 3.4 and Corollary 2.5x, the structure of Demushkin groups, and Pontrjagin’s duality Žcf. wLab 2 , p. 106; Po, Chap. 6x. that there is a quasi-local nonreal field F of zero characteristic, for which the p-component of the character group X Ž GF . is isomorphic to the direct sum of the quasi-cyclic group ZŽ p` . and a cyclic group of order p nŽ p., for each p g P. Let K be a formal power series field in one indeterminate over F. The natural discrete valuation of K is Henselian with a residue field F, so Corollary 4.6 shows that K is stable closed. In view of the Witt isomorphism Br Ž K . ( Br Ž F . [ X Ž GF . wWix, it now remains to be seen that Br Ž F . is a divisible group. Since cdŽ GF . F 2 wCh 2 , Proposition 4.1x, this can be deduced from wSe, Chap. II, Proposition 4x, so Corollary 4.7 is proved. COROLLARY 4.8. Assume that Ž K, ¨ . is a Henselian ¨ alued field satisfying at least one of the conditions Ži. and Žii. of Proposition 3.4. Assume also that ¨ Ž K . is totally indi¨ isible and GK is of cohomological dimension F 2. Then K is stable closed. Proof. The conditions on ¨ Ž K . and GK imply the inequalities 1 F t Ž p . F 2 and cd p Ž GKˆ . F 2 y t Ž p ., for every prime p / char Kˆ Žcf. wCh 2 , Lemma 1.2x.. The assumptions on Ž K, ¨ . and the fact that ¨ Ž K . is totally indivisible indicate that Kˆ is a perfect field of characteristic q G 0, and also, that t Ž q . s 1 in case q ) 0. It is clear from these results and Propositions 3 and 5 of wSe, Chap. IIx, that Br Ž Kˆ1 . s  04 for every finite ˆ Our assertion follows now directly from Proposition extension Kˆ1 of K. 4.3. REFERENCES wAx wAJx

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