2-Galois groups and the Kaplansky radical

An accurate description of the Galois group GF(2) of the maximal Galois 2-extension of a field F may be given for fields F admitting a 2-henselian valuation ring. In this note we generalize this result by characterizing the fields for which GF(2) decomposes as a free pro-2 product F∗H where F is a f...

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Дата:	2010
Автори:	Dario, R.P., Engler, A.J.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154763
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Цитувати:	2-Galois groups and the Kaplansky radical / R.P. Dario, A.J. Engler // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 29–50. — Бібліогр.: 19 назв. — англ.

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spelling	irk-123456789-1547632019-06-16T01:31:49Z 2-Galois groups and the Kaplansky radical Dario, R.P. Engler, A.J. An accurate description of the Galois group GF(2) of the maximal Galois 2-extension of a field F may be given for fields F admitting a 2-henselian valuation ring. In this note we generalize this result by characterizing the fields for which GF(2) decomposes as a free pro-2 product F∗H where F is a free closed subgroup of GF(2) and H is the Galois group of a 2-henselian extension of F. The free product decomposition of GF(2) is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of F. Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application. 2010 Article 2-Galois groups and the Kaplansky radical / R.P. Dario, A.J. Engler // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 29–50. — Бібліогр.: 19 назв. — англ. 2000 Mathematics Subject Classification:12J10; 12F10.. http://dspace.nbuv.gov.ua/handle/123456789/154763 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	An accurate description of the Galois group GF(2) of the maximal Galois 2-extension of a field F may be given for fields F admitting a 2-henselian valuation ring. In this note we generalize this result by characterizing the fields for which GF(2) decomposes as a free pro-2 product F∗H where F is a free closed subgroup of GF(2) and H is the Galois group of a 2-henselian extension of F. The free product decomposition of GF(2) is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of F. Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application.
format	Article
author	Dario, R.P. Engler, A.J.
spellingShingle	Dario, R.P. Engler, A.J. 2-Galois groups and the Kaplansky radical Algebra and Discrete Mathematics
author_facet	Dario, R.P. Engler, A.J.
author_sort	Dario, R.P.
title	2-Galois groups and the Kaplansky radical
title_short	2-Galois groups and the Kaplansky radical
title_full	2-Galois groups and the Kaplansky radical
title_fullStr	2-Galois groups and the Kaplansky radical
title_full_unstemmed	2-Galois groups and the Kaplansky radical
title_sort	2-galois groups and the kaplansky radical
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/154763
citation_txt	2-Galois groups and the Kaplansky radical / R.P. Dario, A.J. Engler // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 29–50. — Бібліогр.: 19 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT dariorp 2galoisgroupsandthekaplanskyradical AT engleraj 2galoisgroupsandthekaplanskyradical
first_indexed	2025-07-14T06:52:19Z
last_indexed	2025-07-14T06:52:19Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 10 (2010). Number 2. pp. 29 – 50 c© Journal “Algebra and Discrete Mathematics” 2-Galois groups and the Kaplansky radical Ronie Peterson Dario, Antonio José Engler Communicated by L. A. Kurdachenko Abstract. An accurate description of the Galois group GF (2) of the maximal Galois 2-extension of a field F may be given for fields F admitting a 2-henselian valuation ring. In this note we generalize this result by characterizing the fields for which GF (2) decomposes as a free pro-2 product F ∗ H where F is a free closed subgroup of GF (2) and H is the Galois group of a 2-henselian exten- sion of F . The free product decomposition of GF (2) is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of F . Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application. Introduction A field F is said to be 2-henselian in case F admits a valuation ring A which extends uniquely to the quadratic closure F (2) of F (F (2) is the inductive limit of the direct system of all Galois extensions of F of finite degree a power of 2 within a fixed algebraically closed field). Henselianity depends on A and it is more appropriate to say that the pair (F,A) is a 2-henselian valued field. For a 2-henselian valued field (F,A) the Galois group GF (2) = Gal(F (2);F ) is well-known, in case the residue class field of A has characteristic 6= 2, [6, Proposition 1.1]. Additionally, if the value group of A is not 2-divisible, then 2-henselian valued fields can be characterized as those fields for which GF (2) has a sufficiently large This work is part of the Ph. D. dissertation of the first author at Campinas State University under the supervision of the second author. Financial support for this research was provided by CAPES. 2000 Mathematics Subject Classification: 12J10; 12F10. Key words and phrases: Brauer group, free pro-2 product, Galois group, 2- henselian valuation ring, quadratic form. 30 2-Galois groups and the Kaplansky radical abelian normal subgroup, [6, Theorem 4.3]. A similar statement holds for absolute henselianity, as it is shown in [12]. This note concerns to the description of GF (2) for valued fields (F,A) which one may consider “almost” 2-henselian as we shall describe now. For a valued field (F,A), where A has residue class field of characteristic 6= 2, it is well-known that A is 2-henselian if and only if 1 +mA ⊂ (F×)2, where mA is the maximal ideal of A and (F×)2 is the subgroup consisting of non-zero squares of the multiplicative group F× = F r { 0 }. To define a pre-2-henselian valuation ring we substitute (F×)2 by the Kaplansky radical R(F ) of F (see Section 1). Thus we call a valuation ring A of F pre-2-henselian if the residue class field kA of A has characteristic 6= 2 and 1 +mA ⊂ R(F ). In the trivial case R(F ) = (F×)2, we get a 2-henselian valuation ring. In general, take an extension A(2) of A to F (2) and write Fh for the decomposition field of A(2) over F [5, §15]. Set Ah = A(2) ∩ Fh. It follows from [5, 15.7] that (Fh, Ah) is a 2-henselian valued field. Moreover, since we are assuming that kA has characteristic 6= 2, Theorem 15.7 of [5] implies that (Fh, Ah) is the smallest 2-henselian valued extension of (F,A) for which A(2) lies over Ah. The pair (Fh, Ah) is then called a 2-henselization of (F,A). A pre-2-henselian valued field (F,A) has many significant properties determined by the corresponding properties of (Fh, Ah). Notably, properties concerning quadratic forms, quaternion algebras, Brauer groups, and orderings. The strong connection between (F,A) and (Fh, Ah) can be characterized by the decomposition GF (2) = F ∗GFh (2) as a free pro-2 product of GFh (2) and a closed free subgroup of GF (2) (see theorems 4.2 and 4.7). Moreover, A can be chosen in a natural way which determines uniquely GFh (2) (up to inner automorphisms of GF (2)). The rank of F is equal to the rank of the quotient group (1 + mA)(F ×)2/(F×)2, which equals the rank of R(F )/(F×)2 in most cases. In the next section we summarize significant properties of R(F ) which are necessary for the paper. The pre-2-henselian properties connecting the objects GF (2), R(F ), and the valuation ring A, are the subject of Section 2. There we also show how to construct pre-2-henselian valuation rings. In Section 3 the connection between the Brauer groups of F and Fh, a 2-henselization of F , is established. We derive from this result a characterization of pre-2-henselian fields by means of its Brauer group which will be necessary to prove, in Section 4, the main results of the paper. In the last section, as an application of theorems 4.2 and 4.7, we show how to produce examples of pre-2-henselian fields whose radical satisfies prescribed conditions. For instance, for any given pair of positive integers m and n, n > 1, there exists a pre-2-henselian field F such that R. P. Dario, A. J. Engler 31 F×/R(F ) and R(F )/(F×)2 have respectively rank m and n. Examples constructed in this way cover previous examples constructed by Berman [2] and Kula [13]. We finish the introduction remarking that all fields considered have characteristic 6= 2, (char 6= 2) and all subgroups of profinite groups are closed. For quadratic forms we use the standard terminology as is found in [14]. For instance given a quadratic form 〈a1, . . . , an〉 over a field F let DF 〈a1, . . . , an〉 be the set of non-zero elements of F represented by 〈a1, . . . , an〉. 1. On Kaplansky radical of a field In a paper from 1969, Kaplansky [10] introduced the radical R(F ) = {a ∈ F× \| 1 ∈ DF 〈a, b〉 for every b ∈ F×} of a field F . Cordes [3] called R(F ) Kaplansky radical of F and rewrite the definition as R(F ) = ⋂ a∈F× DF 〈1, a〉. Alternatively, x ∈ R(F ) if and only if DF 〈1,−x〉 = F×. Clearly R(F ) is a subgroup of F× containing (F×)2. Some further significant properties of R(F ) are: 1.1. R(F ) is additively closed if and only if R(F ) = DF 〈1, 1〉 = ∑ (F×)2 is the set of all finite sums of squares of F×, [14, Corollary 6.5(2), p 452]. Fields for which R(F ) is additively closed are classified in two types: 1.2. If F is non-real, i.e., −1 ∈ ∑ (F×)2, then R(F ) = F×. 1.3. In the formally real case, R(F ) is a preordering (preorderings can be found in [14, p 289]). 1.4. (F× : R(F )) = 2 if and only if F is formally real, uniquely ordered by R(F ), and there exists a unique, up to isomorphism, quaternion di- vision algebra over F , [14, Theorem 6.10, p 454]. In this case Lam [14, Definition 6.7, p 453] called F a pre-Hilbertian field. 2. Pre-2-henselianity Fields admitting a proper pre-2-henselian valuation ring will be obtained by applying to the subgroup R(F ) the methods developed in [1]. To this end, let us first define R(F )-rigid elements as being those a ∈ F×r±R(F ) for which {x + ay 6= 0 \| x, y ∈ R(F ) ∪ { 0 }} = R(F ) ∪ aR(F ). If x and 32 2-Galois groups and the Kaplansky radical −x are both R(F )-rigid, then we say that x (and −x too) is R(F )-birigid. Non-R(F )-birigid elements also play an important role in constructing pre-2-henselian valuation rings. Denote B(R(F )) = {x ∈ F× \| x is non-R(F)-birigid }. The elements of B(R(F )) are called basics. A surpris- ingly fact is that B(R(F )) is a subgroup of F×. In the next result we formalize properties of R(F )-rigid and basic elements which will be crucial to establish the existence of a pre-2-henselian valuation ring. Lemma 2.1. Let F be a field and a ∈ F× r±R(F ). Then: (1) a is R(F )-rigid if and only if DF 〈1, a〉 = R(F ) ∪ aR(F ). (2) B(R(F )) is a subgroup of F×. Proof. For a quadratic form 〈1, a〉 it follows from [3, Proposition 1] that DF 〈1, a〉 = {x + ay 6= 0 \| x, y ∈ R(F ) ∪ { 0 }}; the equality implies (1). Also it implies that {x+ay 6= 0 \| x, y ∈ R(F )∪{ 0 }} is a subgroup of F×. Therefore, by [19, Proposition 2.4], B(R(F )) is a subgroup of F×. In the trivial case R(F ) = (F×)2 we get the usual rigid elements (or (F×)2-rigid). We know that 2-henselian valuation rings arise from the existence of enough rigid elements (see [1] or [19]). Pre-2-henselian valuation rings are obtained by replacing rigid elements by R(F )-rigid elements in the usual procedures. Theorem 2.2. Let F be a field such that B(R(F )) 6= F×. Then at least one of the following conditions holds: (1) F admits a proper pre-2-henselian valuation ring A with non-2- divisible value group ΓA. (2) F is formally real with at most two orderings and R(F ) = ∑ (F×)2. Consequently (F× : R(F )) ≤ 4. The proof of Theorem 2.2 is partially based on Theorem 3.9 from [1] which requires the characterization of the “exceptional case,” according to the designation given by Arason et al. [1, Definition 2.15]. We say that R(F ) is exceptional if B(R(F )) = R(F ) ∪ −R(F ) and either −1 ∈ R(F ) or R(F ) is additively closed. If R(F ) + R(F ) ⊂ R(F ) and F is formally real we know from (1.3) that R(F ) is a preordering. If additionally R(F ) is exceptional, then R(F ) will possess a new interesting property, namely R(F ) will be a fan. See [15, §5] for more about fans. In the non-real case, if R(F ) is exceptional, then B(R(F )) = R(F ). Thus assertion (1.4) implies either B(R(F )) = F× or (F× : B(R(F )) > 2. For further reference let us remark the above two cases: R. P. Dario, A. J. Engler 33 Lemma 2.3. Assume R(F ) 6= F× is exceptional. (1) If F is non-real, then (F× : B(R(F ))) > 2. (2) If F is formally real, then R(F ) is a fan. The basic connection between R(F ) and valuation rings of F is given in the next proposition. For a valuation ring A of F , let A× = ArmA, πA : A → kA, ΓA, and vA be respectively the group of units, the canonical homomorphism, the value group and a valuation corresponding to A. Proposition 2.4. For a field F with radical R(F ) 6= F× let A be a valuation ring of F . (1) πA(R(F ) ∩A×) ⊂ R(kA). (2) If A is pre-2-henselian and ΓA = 2ΓA, then πA(R(F )∩A×) = R(kA). Additionally R(kA) 6= k×A . (3) If R(F ) 6⊂ (1 +mA)(F ×)2, then (ΓA : 2ΓA) ≤ 2. Moreover, (ΓA : 2ΓA) = 2 implies k×A = (k×A) 2. (4) If (1 +mA)(F ×)2 & R(F ), then ΓA = 2ΓA. Proof. (1) For u ∈ (1 +mA)(F ×)2 ∩A×, πA(u) ∈ (k×A) 2 ⊂ R(kA). Take then u ∈ R(F ) ∩A× and u 6∈ (1 +mA)(F ×)2. For any z ∈ A× there are a, b ∈ (F×)2 such that z = a− ub (since u ∈ R(F )). From vA(a− ub) = 0 one gets vA(a), vA(ub) ≥ 0. Otherwise, vA(a) < 0 or vA(ub) = vA(b) < 0 would imply vA(a) = vA(b) (< 0) 1. Then u = (b−1a)(1 − za−1) ∈ (1 + mA)(F ×)2 contradicting the choice of u. Hence πA(z) = πA(a) − πA(u)πA(b) ∈ DkA〈1,−πA(u)〉. Thus k×A = DkA〈1,−πA(u)〉 which implies πA(u) ∈ R(kA), as required. (2) Due to the previous item (1) it remains to show the inclusion R(kA) ⊂ πA(R(F ) ∩ A×) to prove the equality. For u ∈ A× such that πA(u) ∈ R(kA) we have that DkA〈1,−πA(u)〉 = k×A . Therefore given z ∈ A× there exist a, b ∈ (A×)2 such that πA(z) = πA(a)− πA(u)πA(b). Thus z(a− ub)−1 ∈ 1 +mA ⊂ R(F ). Since R(F ) ⊂ DF 〈1,−u〉 it follows that z ∈ DF 〈1,−u〉. Hence A× ⊂ DF 〈1,−u〉. The assumption ΓA = 2ΓA will then imply F× = A×(F×)2 ⊂ DF 〈1,−u〉. Consequently u ∈ R(F ) and πA(u) ∈ πA(R(F ) ∩A×), as required. Finally, R(kA) = k×A would imply πA(R(F ) ∩ A×) = πA(A ×) and so A× ⊂ R(F ). Thus F× = A×(F×)2 = R(F ), a contradiction. (3) Take r ∈ R(F ) r (1 + mA)(F ×)2 and a, b ∈ (F×)2. If vA(a) = vA(rb) < vA(a−rb), then vA(1−rba−1) > 0. Consequently rba−1 ∈ 1+mA, 1vA(a) 6= vA(rb) implies vA(a− rb) = min{vA(a), vA(rb)} 34 2-Galois groups and the Kaplansky radical a contradiction. Thus vA(a−rb) = min{vA(a), vA(rb)}, for all a, b ∈ (F×)2. As F× = DF 〈1,−r〉, vA(x) ∈ 2ΓA ∪ (v(r) + 2ΓA), for every x ∈ F×. Thus ΓA ⊂ 2ΓA ∪ (v(r) + 2ΓA) (1) and so (ΓA : 2ΓA) ≤ 2 as desired. We next prove that (ΓA : 2ΓA) = 2 implies k×A = (k×A) 2. In fact, for r ∈ R(F ) r (1 +mA)(F ×)2, as above, vA(r) 6∈ 2ΓA by equation (1). Moreover, for y ∈ A× and a, b ∈ (F×)2 satisfying y = a − rb we have seen that 0 = vA(y) = min{vA(a), vA(rb)}. The case vA(rb) = 0 cannot occur, otherwise vA(r) = −vA(b) ∈ 2ΓA. Consequently vA(rb) > 0 and vA(a) = 0. Hence a ∈ (A×)2 and πA(y) = πA(a) ∈ (k×A) 2, as required. (4) Assuming ΓA 6= 2ΓA, it follows from the previous item (3) that (ΓA : 2ΓA) = 2 and k×A = (k×A) 2. The last equality implies A× = (1+mA)(A ×)2. Thus A× ⊂ R(F ) (†). From (ΓA : 2ΓA) = 2 and equation (1) we can write F× = A×(F×)2 ∪ rA×(F×)2 (‡), for any r ∈ R(F )r (1 +mA)(F ×)2. Combining (†) and (‡) we get a contradiction. Corollary 2.5. Keep the notation of Proposition 2.4 and let A be a pre-2- henselian valuation ring of F . If ΓA 6= 2ΓA, then (1 +mA)(F ×)2 = R(F ). We now present the proof of Theorem 2.2. Proof of Theorem 2.2. We shall consider two cases. The first case corre- sponds to R(F ) non-exceptional or R(F ) exceptional and F is non-real. The other case corresponds to R(F ) exceptional and F is formally real. In the first case, [1, Theorem 3.9] guarantees the existence of a valuation ring A, pre-2-henselian, such that A×R(F ) = B(R(F )) if R(F ) is not exceptional and (A×R(F ) : B(R(F ))) ≤ 2, otherwise. Particularly, if R(F ) is exceptional and F is non-real, then (F× : A×R(F )) ≥ 2, by Lemma 2.3 (1). Therefore, in both cases, F× 6= A×R(F ) and A is a proper subring of F . Moreover, since A×(F×)2 ⊂ A×R(F ) we also get F× 6= A×(F×)2 and so ΓA 6= 2ΓA. To complete the proof in this case it remains to be seen that char kA 6= 2, i.e., the first case implies (1) of Theorem 2.2. By [1, Lemma 4.4], char kA 6= 2 follows from (F× : R(F )) > 2. In the non-exceptional case, as R(F ) & B(R(F )) and B(R(F )) 6= F× (by assumption), (F× : R(F )) > 2 holds. In the non-real exceptional case, Lemma 2.3 (1) implies (F× : R(F )) > 2 because R(F ) ⊂ B(R(F )). Assume finally that R(F ) is exceptional and F is formally real (i.e., B(R(F )) = ±R(F ), −1 6∈ R(F ), and R(F ) + R(F ) ⊂ R(F )). Then Lemma 2.3 (2) shows that R(F ) is a fan. By Bröcker’s Trivialization R. P. Dario, A. J. Engler 35 Theorem [15, Theorem 5.13], F admits a valuation ring A which is pre- 2-henselian and such that πA(R(F )) is a trivial fan, i.e., a preordering such that (k×A : πA(R(F ))) ≤ 4. Since kA is a formally real field we get char kA 6= 2. If ΓA 6= 2ΓA we get again the conclusion (1) of Theorem 2.2. Next, we assume that ΓA is 2-divisible and prove that the conclusion (2) of Theorem 2.2 holds. For ΓA = 2ΓA, Proposition 2.4 (2) implies R(kA) = πA(R(F )), which we know to be additively closed. Apply- ing now assertion (1.1) to both radicals we get R(F ) = ∑ (F×)2 and R(kA) = ∑ (k×A) 2. The last equality, R(kA) = ∑ (k×A) 2, combined with (kA : R(kA)) ≤ 4 implies that kA has at most two orderings. On the other hand, the equality R(F ) = ∑ (F×)2 implies that A is compatible with every ordering of F [15, Definition 2.4]. Consequently, by [15, Corollary 3.11], the orderings of F correspond bijectively with the orderings of kA if we keep in mind that ΓA = 2ΓA. Hence F has at most two orderings and (F× : R(F )) ≤ 4. In case (2) of Theorem 2.2, R(F ) = ∑ (F×)2 is a trivial fan and fields for which ∑ (F×)2 is a trivial fan are well-know. Moreover, such a field F may or may not admit a pre-2-henselian valuation ring with non-2-divisible value group. For instance, if F admits an Archimedean ordering, then no valuation ring could be pre-2-henselian. On the other hand, the existence of a pre-2-henselian valuation ring with non-2-divisible value group can only occur if (ΓA : 2ΓA) = 2 and kA is uniquely ordered, by [15, Corollary 3.11]. The ordering of kA is given by R(kA) and so kA is a pre-Hilbertian field (see (1.4)). We shall now investigate the relations between various pre-2-henselian valuation rings of a field. Like for 2-henselian valuation rings we shall see for a field F 6= F (2) that any two pre-2-henselian valuation rings C and D are comparable provided that kC or kD is non-quadratically closed2. Moreover if exactly one of them, say kD, is quadratically closed, then D ⊂ C. In order to describe the set of all pre-2-henselian valuation rings of F we set up some notations. We denote by Ω1 the set of all pre-2-henselian valuation rings of F with non-quadratically closed residue class field and by Ω2 the set of all valuation rings having residue class field quadratically closed. Set Ω = Ω1 ∪Ω2. Before we describe the properties of Ω1 and Ω2 let us recall, for any two valuation rings C and D, that C is said to be coarser than D (or D is said to be finer than C) if D ⊂ C. 2Since we are assuming char 6= 2 for every field there is no distinction between quadratically closed and quadratic separably closed. 36 2-Galois groups and the Kaplansky radical Proposition 2.6. Keep notations as above and assume that R(F ) 6= F×. (1) If a valuation ring C of F is coarser than A ∈ Ω, then C ∈ Ω. (2) Ω1 is totally ordered (by inclusion) and closed by the coarsening relation. (3) If A ∈ Ω2 is coarser than a valuation ring C of F , then C ∈ Ω2. Moreover, for C, D ∈ Ω2, their compositum C ·D lies in Ω2 as well. (4) For any pair C ∈ Ω2 and A ∈ Ω1, A is coarser than C. (5) If Ω2 6= ∅, then for every A ∈ Ω2, ΓA 6= 2ΓA and (1+mA)(F ×)2 = R(F ). We need the following two lemmas to prove the proposition. Lemma 2.7. For a field F let T be a subgroup of F× and let A and C be independent valuation rings of F , i.e., A · C = F . If 1 +mA ⊂ T and 1 +mC ⊂ T , then T = F×. Proof. By the Approximation Theorem [7, Theorem 2.4.1] for every x ∈ F× there exists y ∈ F× such that vC(y − x) > vC(x) and vD(y − 1) > 0. Thus yx−1 ∈ 1 +mC and y ∈ 1 +mD. Hence x = y(yx−1)−1 ∈ T . Lemma 2.8. Let A and C be incomparable (by inclusion) pre-2-henselian valuation rings of a field F . Write D = A · C, for their compositum and assume that R(F ) 6= F×. Then A, C, D ∈ Ω2. Proof. From A ⊂ D it follows that mD ⊂ mA. Thus D is also pre-2- henselian. Going to residue class field of D we set R(F ) = πD(R(F ) ∩ D×), A = πD(A), and C = πD(C). Clearly 1 + mA ⊂ R(F ) and 1 + mC ⊂ R(F ) hold. On the other hand A and C are independent valuation rings of kD, by construction. Applying Lemma 2.7 to them one gets R(F ) = k×D. By Proposition 2.4 (1) R(F ) ⊂ R(kD). Therefore R(kD) = k×D. On the other hand, under the assumption (1 + mD)(F ×)2 & R(F ) we would get from Proposition 2.4 (4) that ΓD = 2ΓD. But this would contradict item (2) of Proposition 2.4. Therefore (1 +mD)(F×)2 = R(F ). Applying now the conclusions of the previous paragraph we get k×D = R(F ) = (k×D) 2. Thus D ∈ Ω2. Finally since A and A have the same residue class field, also kA is quadratically closed. Hence A ∈ Ω2 follows from A ⊂ D. Similarly C ∈ Ω2. We are now ready to prove Proposition 2.6. R. P. Dario, A. J. Engler 37 Proof of Proposition 2.6. (1) and the first part of (3) are consequences of general valuation theory; (2), the last part of (3), and (4) follow directly from the above Lemma 2.8. To prove (5), observe first that if A ∈ Ω2 then A× = (1 + mA)(A ×)2. Under the condition ΓA = 2ΓA, it follows that F× = A×(F×)2 = (1 +mA)(F ×)2 ⊂ R(F ) ⊂ F×. Thus F× = R(F ), contrary to the assumption. Hence ΓA 6= 2ΓA, for every A ∈ Ω2. Finally, by Corollary 2.5, (1 +mA)(F ×)2 = R(F ). Observe that F ∈ Ω is always true and so Ω 6= ∅. For F 6= F (2) we cannot say Ω2 6= ∅, however. In Section 4 we shall see that the existence of A ∈ Ω, A 6= F , is the main requirement to describe GF (2). Remark 2.9. For further use, we end this section remarking that (Fh) × = F×(F× h )2 and (1 +mA)(F ×)2 = (F× h )2 ∩ F× hold for a 2-henselization (Fh, Ah) of a valued field (F,A). They are consequence of two facts: (Fh, Ah) is an immediate extension of (F,A) [5, Theorem 15.8], and 1 + mAh ⊂ (F× h )2. 3. Brauer group of pre-2-henselian valued fields This section is dedicated to study the relative Brauer group 2Br(F ) consist- ing of all classes of central simple F -algebras split by F (2). Setting Br(F ) for the Brauer group of F we may describe 2Br(F ) as 2Br(F ) = { z ∈ Br(F ) \| 2z = 0 }. In 1981 Merkurjev [16] proved the long standing conjec- ture that 2Br(F ) is generated by the set of quaternion algebras (F ; a, b) defined over F . Recall that for a, b ∈ F× the quaternion algebra (F ; a, b) is the F -algebra having generators i, j and relations i2 = a, j2 = b and ij = −ji (see for example [14, Chapter III, p 51]). Quaternion algebras over F are connected with the Kaplansky radical of F as follows: the correspon- dence from F×/(F×)2×F×/(F×)2 to the Brauer group Br(F ) of F which assigns to each pair (a(F×)2, b(F×)2) the class of the quaternion algebra (F ; a, b) in Br(F ) is a symmetric bimultiplicative pairing. The radical of this pairing consisting of {x ∈ F× \| (F ;x, b) splits for every b ∈ F×} is the Kaplansky radical R(F ) of F , [14, Proposition 6.1]. For a 2-henselization (Fh, Ah) of (F,A) let Res : 2Br(F ) → 2Br(Fh) be the restriction map which assigns to the class [S] of a central simple F -algebra S the class [S ⊗F Fh] ∈ 2Br(Fh). Observe that ker(Res) = Br(Fh;F ) is the relative Brauer group associated with the extension Fh\|F . Our first aim is to prove Theorem 3.2 which gives a description of ker(Res) for an arbitrary field F (charF 6= 2). The main goal in this section follows from this result; a characterization of a pre-2-henselian valued field F by means of 2Br(F ) (Corollary 3.4). Corollary 3.4, which has its own interest, 38 2-Galois groups and the Kaplansky radical provides one of the tools necessary to prove the main theorems of the paper (results 4.2 and 4.7). We start with a technical lemma. Recall first that 〈a1, . . . , an〉 repre- sents the diagonal quadratic form with coefficients a1, . . . , an in a field F . Also the n-fold Pfister form ⊗n i=1〈1,−ai〉 will be denoted by 〈〈−a1,. . ., −an〉〉. As usual W (F ) stands for the Witt ring of all isometric classes of anisotropic quadratic forms over F . Following Lam [14], IF denotes the ideal consisting of classes of even-dimensional quadratic forms and InF is the n-power of IF . All unexplained notations can be found in [14]. Lemma 3.1. Let F be a field and a1, . . . , an ∈ F×, for some n ≥ 2. Then there exists qn ∈ I3F such that 〈〈−a1 · · · an〉〉 = n ∑ i=1 〈〈−ai〉〉 − ∑ 1≤i<j≤n 〈〈−ai,−aj〉〉 + qn. Proof. We argue by induction on n. For n = 2, 〈〈−a1〉〉+ 〈〈−a2〉〉 = 〈〈−a1,−a2〉〉+ 〈〈−a1a2〉〉. Next, for n > 2, 〈〈−a1 · · · an〉〉 = 〈〈−a1 · · · an−1〉〉+ 〈〈−an〉〉 − 〈〈−an〉〉〈〈−a1 · · · an−1〉〉. Applying the induction hypothesis to 〈〈−a1 · · · an−1〉〉 one gets qn−1 ∈ I3F such that 〈〈−a1 · · · an〉〉 can be written as n−1 ∑ i=1 〈〈−ai〉〉− ∑ 1≤i<j≤n−1 〈〈−ai,−aj〉〉−〈〈−an〉〉 ( n−1 ∑ i=1 〈〈−ai〉〉− ∑ 1≤i<j≤n−1 〈〈−ai,−aj〉〉 ) + 〈〈−an〉〉+ qn−1 − 〈〈−an〉〉qn−1 = = n ∑ i=1 〈〈−ai〉〉− ∑ 1≤i<j≤n 〈〈−ai,−aj〉〉+ ∑ 1≤i<j≤n−1 〈〈−an,−ai,−aj〉〉+qn-1−〈〈−an〉〉qn-1 Setting qn = ∑ 1≤i<j≤n−1 〈〈−an,−ai,−aj〉〉+ qn−1 − 〈〈−an〉〉qn−1, one gets the desired decomposition for 〈〈−a1 · · · an〉〉. For the next theorem let us recall that the map 〈〈−a,−b〉〉 7→ (F ; a, b) extends to a group homomorphism γ : I2F/I3F → 2Br(F ) which, in light of Merkurjev’s Theorem [16], is an isomorphism. Moreover, this is R. P. Dario, A. J. Engler 39 a functorial isomorphism. In particular, the extension Fh\|F gives rise to the following commutative diagram I2F/I3F r−−−−→ I2Fh/I 3Fh γ   y ≀ ≀   y γh 2Br(F ) Res−−−−→ 2Br(Fh) (2) where r is induced by F ⊂ Fh as follows: the inclusion induces naturally a ring homomorphism r : W (F ) → W (Fh). Thus r(InF ) ⊂ InFh, for every n ≥ 0. Actually, r(InF ) = InFh; just observe that r is a surjective map by Remark 2.9 and IF is a maximal ideal. Therefore, r is the quotient homomorphism constructed from r. Note that r is also a surjective map and consequently so is Res. Theorem 3.2. For a valued field (F,A) let (Fh, Ah) be a 2-henselization. Then Br(Fh;F ), the kernel of Res, is generated by the set { (F, a, t) \| a ∈ F×, t ∈ 1 +mA } . Proof. Let us first check that ker(r) is generated by the set { 〈b〉〈〈−a,−t〉〉+ I3F \| a, b ∈ F×, and t ∈ 1 +mA } . (3) We know from [11, Proposition 2.4] that ker(r) is generated by the set { 〈〈−t〉〉 \| t ∈ 1 +mA }. Observe next that for every x+ I3F ∈ ker(r) we may choose x ∈ ker(r) ∩ I2F .3 Hence we may write x = n ∑ i=1 〈ai〉〈〈−ti〉〉, for some ai ∈ F× and ti ∈ 1 +mA. The quadratic form has determinant d(x) = (−1)nt1 · · · tn(F×)2 and dimension 2n. However, according to [14, Corollary 2.2, p 32] d(x) = (−1)n(2n−1)(F×)2, for x ∈ I2F . Combining both facts we obtain t1 · · · tn ∈ (F×)2. By replacing tn with t1 · · · tn−1 in the description of x and then applying Lemma 3.1 we deduce the following 3For x ∈ I2F and y ∈ I3F such that r(x) = r(y) it follows that z = x− y ∈ ker(r). Then z ∈ I2F and z − x = −y ∈ I3F . Hence x+ I3F = z + I3F . 40 2-Galois groups and the Kaplansky radical equations: x = n−1 ∑ i=1 〈ai〉〈〈−ti〉〉+ 〈〈−t1t2 . . . tn−1〉〉〈an〉 = n−1 ∑ i=1 〈ai〉〈〈−ti〉〉+ ( n−1 ∑ i=1 〈〈−ti〉〉 − ∑ 1≤i<j≤n−1 〈〈−ti,−tj〉〉+ qn−1 ) 〈an〉, for some qn−1 ∈ I3F = n−1 ∑ i=1 〈ai, an〉〈〈−ti〉〉 − ∑ 1≤i<j≤n−1 〈〈−ti,−tj〉〉〈an〉+ 〈an〉qn−1, for some qn−1 ∈ I3F. Since each of the above summands belongs to I2F the equation below makes sense in I2F/I3F x+ I3F = n−1 ∑ i=1 〈ai, an〉〈〈−ti〉〉 − ∑ 1≤i<j≤n−1 〈〈−ti,−tj〉〉〈an〉+ I3F. Finally, it follows from 〈ai, an〉〈〈−ti〉〉+ I3F = −〈〈aian,−ti〉〉+ I3F that the set displayed in (3) generates ker(r), as announced. To finish the proof observe that the commutative diagram (2) implies that ker(Res) = γ(ker(r)) and [14, Lemma 3.2, p 114] yields γ(〈b〉〈〈−a, −t〉〉+ I3F ) = (F, a, t), for every a ∈ F×, and t ∈ 1 +mA. Corollary 3.3. Let (F,A) be a valued field and fix a 2-henselization (Fh, Ah). Then 2Br(F ) ∼= 〈 { (F, a, t) \| a ∈ F×, t ∈ 1 +mA } 〉 ⊕ 2Br(Fh), where 〈 {· · · } 〉 stands for the subgroup (of 2Br(F )) generated by the set {· · · }. Proof. The surjectivity of Res implies the exactness of the sequence 0 → Br(Fh;F ) −→ 2Br(F ) Res−→ 2Br(Fh) → 0 which splits when considered as a sequence of F2-vector spaces. A cohomological characterization of pre-2-henselian fields follows im- mediately from Corollary 3.3. Corollary 3.4. For a valued field (F,A) let (Fh, Ah) be a 2-henselization. Then, (F,A) is pre-2-henselian if and only if Res : 2Br(F ) → 2Br(Fh) is an isomorphism. R. P. Dario, A. J. Engler 41 4. Galois characterization of pre-2-henselian fields In this section we give a characterization of a pre-2-henselian field (F,A) by means of the Galois group GF (2). The instrument is the cohomological characterization of free products of pro-p groups established by Neukirch [17, Sätze 4.2, 4.3]. A convenient form of Neukirch’s criterium, translated to the case under consideration in the paper, is as follows: Theorem 4.1. Let F be a field and K1, K2 be two extensions of F in F (2). Then GF (2) ∼= GK1 (2) ∗GK2 (2) if and only if (1) the canonical map res1 : F×/(F×)2 → K× 1 /(K× 1 )2 ×K× 2 /(K× 2 )2 induced by the inclusions is an isomorphism; (2) the canonical map res2 : 2Br(F ) → 2Br(K1)× 2Br(K2) induced by Res : 2Br(F ) → 2Br(K1) and Res : 2Br(F ) → 2Br(K2) is a monomorphism. With the help of Corollary 3.4 and Theorem 4.1 we obtain a Galois characterization of those fields F for which there exists A ∈ Ω such that ΓA 6= 2ΓA. Theorem 4.2. Let (F,A) be a pre-2-henselian valued field with ΓA 6= 2ΓA. Then GF (2) admits a decomposition as free pro-2 product GF (2) = F ∗H of closed subgroups F and H such that (i) F is a free pro-2 group; (ii) H has a non-trivial abelian normal closed subgroup. Moreover the subgroups F , H are naturally associated with R(F ) and A: (I) rank (F) = rank ( R(F )/(F×)2 ) . (II) Set µ = rank ( ΓA/2ΓA ) . Then H = Zµ 2 ⋊GkA(2) 4. Conversely, let F be a field such that GF (2) = F ∗ H where (a) F is a free pro-2 group; 4The action of GkA (2) on the abelian component is described in [6, Proposition 1.1]. 42 2-Galois groups and the Kaplansky radical (b) rank (H) ≥ 2, H 6∼= Z2 ⋊ Z/2Z, and H has a non-trivial abelian normal closed subgroup. Then F admits a pre-2-henselian valuation ring A with ΓA 6= 2ΓA. Finally, the decomposition GF (2) = F ∗ H does not depend on the valuation ring A and H is uniquely determined (up to inner automorphisms of GF (2)). The proof of Theorem 4.2 depends essentially on the next three results which also have their own interest. We start by characterizing those fields F such that GF (2) is a free pro-2 group by means of R(F ). Proposition 4.3. For a field F the following conditions are equivalent: (1) R(F ) = F×. (2) 2Br(F ) = 0. (3) GF (2) is a free pro-2 group. Proof. The equivalence (1) ⇔ (2) is clear if we observe that R(F ) = F× occurs if and only if (F ; a, b) split for all a, b ∈ F× which turns to be equivalent to 2Br(F ) = 0, by Merkurjev’s Theorem [16]. The proof of (2) ⇔ (3) follows from [18, Corollary 3.8, p 262, and Corollary 3.2, p 255]. Proposition 4.4. For a pre-2-henselian valued field (F,A), A 6= F , fix a 2-henselization (Fh, Ah). Then there exists an extension K ⊂ F (2) satisfying the following conditions: (1) R(K) = K× and the inclusion F ⊂ K induces an isomorphism K×/(K×)2 ∼= (1 +mA)(F ×)2/(F×)2. (2) GF (2) = GK(2) ∗GFh (2). Proof. Take an extension K ⊂ F (2) of F such that the natural map F×/(F×)2 → K×/(K×)2 restricts to an injective map on group (1 + mA)(F ×)2/(F×)2 and K is maximal with this property. We claim this map induces an isomorphism (1+mA)(F ×)2/(F×)2 ∼= K×/(K×)2. In fact, the map is injective by construction and what remains to be proved is the surjectivity. Take any x ∈ K× and assume, for the sake of contradiction, that x(K×)2 is not in the image of (1 + mA)(F ×)2/(F×)2. Form the quadratic extension L = K( √ x). Since (L×)2 ∩ K = (K×)2 ∪ x(K×)2 it follows that (1 +mA)(F ×)2/(F×)2 is sent injectively into L×/(L×)2, R. P. Dario, A. J. Engler 43 contradicting the maximality of K. The last argument can also be displayed as K× = (1 +mA)(K ×)2. (4) Thus if 1 + mA ⊂ R(K), then K× = R(K), as desired. However, if x ∈ 1 + mA ⊂ R(F ), then F× ⊂ DF 〈1,−x〉 ⊂ DK〈1,−x〉. Hence 1 + mA ⊂ DK〈1,−x〉 which combined with the above equation (4) implies K ⊂ DK〈1,−x〉. So 1 +mA ⊂ R(K) already occurs. To finish the proof it remains to show that item (2) of Proposition 4.4 also holds. To prove the decomposition GF (2) = GK(2) ∗GFh (2) we have to show that the conditions (1) and (2) from Theorem 4.1 are fulfilled. • res1 is injective. Assume x ∈ (K×)2 and x ∈ (F× h )2 for some x ∈ F×. According to Remark 2.9, (F× h )2 ∩ F = (1 + mA)(F ×)2. On the other hand, it follows from the very construction of K that (K×)2 ∩ (1 +mA)(F ×)2 = (F×)2. Consequently x ∈ (F×)2, proving that res1 is injective. • res1 is surjective. Let x ∈ K× and y ∈ F× h , be arbitrarily chosen. From Remark 2.9 there exists w ∈ F× such that w−1y ∈ (F× h )2. Next, by applying the above condition (4) to w−1x ∈ K× we get u ∈ (1+mA) such that u−1w−1x ∈ (K×)2. Since 1 +mA ⊂ (F× h )2 we also have u−1w−1y ∈ (F× h )2. Therefore, putting the things together we get res1(wu(F×)2) = ( x(K×)2, y(F× h )2 ) , proving the required surjectivity. • res2 is injective. R(K) = K× was obtained at the beginning of the proof. Thus, according to Proposition 4.3, 2Br(K) = 0. Therefore res2 = Res, which is an isomorphism by Corollary 3.4. Proposition 4.5. Let F be a field with R(F ) 6= F×. Let K, H ⊂ F (2) be two extensions of F such that GK(2) is free and H admits a 2-henselian valuation ring C 6= H. If GF (2) = GK(2) ∗ GH(2), then A = C ∩ F is a pre-2-henselian valuation ring of F . If ΓA is not 2-divisible, then this decomposition implies that (H,C) is a 2-henselization of (F,A). In the case ΓA = 2ΓA, (H,C) contain a 2-henselization (Fh, Ah) of (F,A) such that GF (2) = GK1 (2) ∗ GFh (2), for a 2-extension F ⊂ K1 ⊂ F (2) with GK1 (2) free. Proof. Let us call R = (H×)2 ∩ F×. We first prove that R ⊂ R(F ). For r ∈ R we shall show that (F ; r, x) splits for every x ∈ F×. In fact, given x ∈ F×, we have that (H; r, x) splits, because r ∈ (H×)2. On the other hand, by Proposition 4.3, 2Br(K) = 0. So (K; r, x) also splits. Consequently, in the notation of Theorem 4.1, the injectivity of res2 implies that (F ; r, x) splits, as required. Thus R ⊂ R(F ), as claimed. In the next step, observe that 1+mA ⊂ 1+mC ⊂ (H×)2 implies that 1 +mA ⊂ R and so (F,A) is pre-2-henselian, as required. 44 2-Galois groups and the Kaplansky radical Now we prove the final part of the proposition. Recall that the valued field (H,C) must contain a 2-henselization (Fh, Ah) of (F,A), because (H,C) is a 2-henselian extension of (F,A). Since Fh ⊂ H are two subex- tensions of F (2), we have either Fh = H or there exists an intermediate extension Fh ⊂ L ⊂ H such that [L : Fh] = 2. If the last case occurs, there exists x ∈ Fh r (F× h )2 such that x ∈ (H×)2. According to Remark 2.9, we may take x ∈ F . By the first part of the proof we have x ∈ R(F ). Since x 6∈ (F× h )2 also x /∈ (1+mA)(F ×)2. Then Corollary 2.5 implies ΓA = 2ΓA. Thus the inequality F 6= H only occurs if ΓA = 2ΓA. In this case, we apply the Proposition 4.4 to get the desired decomposition. Now that the connection between free products of suitable subgroups of GF (2) and pre-2-henselian valuation rings was established the proof of our first main result is just a matter of a proper combination of all ingredients. Proof of Theorem 4.2. Proposition 4.4 gives us the free product decom- position GF (2) = GK(2) ∗ GFh (2) for suitable extensions K, Fh ⊂ F (2) of F . Let us check that F = GK(2) and H = GFh (2) have the properties announced in (i), (ii), (I) and (II). That F = GK(2) is a free pro-2 group follows from Proposition 4.3. From ΓA 6= 2ΓA we deduce by means of Corollary 2.5 that R(F ) = (1 + mA)(F ×)2 6= F×. Hence K×/(K×)2 ∼= R(F )/(F×)2, by Proposition 4.4. From [18, Proposition 6.2, Theorem 6.8] and Galois Theory we have that rank (F) is equal to rank (K×/(K×)2). Thus (I) holds. The existence of a non-trivial abelian normal subgroup of GFh (2) follows from valuation theory. In fact, a proof of it, as well as a proof of (II), can be found in § 1 of [6] during the demonstration of Proposition 1.1. Conversely, assume GF (2) ∼= F∗H as stated in the theorem. Let K and H be respectively the fixed fields of F and H. We now use Proposition 4.5. We already know that GK(2) is free. By applying [6, Theorem 4.3] to the Galois extension F (2)\|H in the case p = 2 we obtain that H carries a 2-henselian valuation ring C with value group Γ 6= 2Γ and residue class field k such that char k 6= 2. Thus C 6= H, as desired. Then A = C ∩ F is pre-2-henselian and ΓA 6= 2ΓA, because the same is true for Γ. Hence (H,C) is a 2-henselization of (F,A) (Proposition 4.5). The last statement of the theorem says that the decomposition does not depend on A and H is unique. To prove it we write a preparatory lemma. For any valuation ring C of F (2) let Gh(C) stand for the decomposition group of C over F [5, §15]. R. P. Dario, A. J. Engler 45 Lemma 4.6. For a field F assume there are A, C ∈ Ω with non-2-divisible value group. Then we can choose extensions A(2) and C(2) of A and C to F (2), respectively, such that Gh(A(2)) = Gh(C(2)). The lemma above enables us to finish the proof of Theorem 4.2. We have already proved that any decomposition GF (2) = F1 ∗ H1 which satisfies the assumptions (i) and (ii) of Theorem 4.2 yields C ∈ Ω such that ΓC 6= 2ΓC and H1 = Gh(C(2)), for some extension C(2) of C to F (2). Consequently, Lemma 4.6 above says that H1 does not depend on a particular choice of C ∈ Ω with non 2-divisible value group. Set next GF (2) = F ∗ H and GF (2) = F1 ∗ H1 where H = Gh(A(2)) and H1 = Gh(C(2)) for valuation rings A(2) and C(2) of F (2) lying, respectively, over two valuation rings A, C ∈ Ω of F , with non 2-divisible value groups. By Lemma 4.6 we can choose A(2)′ and C(2)′, respectively the exten- sions of A and C to F (2) such that Gh(A(2)′) = Gh(C(2)′). By the Conjugation Theorem [7, 3.2.15] there exist g, h ∈ GF (2) such that A(2)′ = g(A(2)) and C(2)′ = h(C(2)). We derive from these equali- ties Gh(A(2)′) = gGh(A(2))g−1 and Gh(C(2)′) = hGh(C(2))h−1. Thus H = g−1hH1h −1g, are conjugated. Proof of Lemma 4.6. Assume first that A and C are comparable by inclu- sion; say C is finer than A. Take an extension C(2) of C to F (2) finer than A(2), an extension of A to F (2). We shall show that Gh(A(2)) = Gh(C(2)), and additionally, this equality implies that C(2) is the unique extension of C to F (2) finer than A(2). Let F h(C(2)) and F h(A(2)) be the fixed fields of Gh(C(2)) and Gh(A(2)), respectively. From C(2) ⊂ A(2) one gets Gh(C(2)) ⊂ Gh(A(2)) and so F h(A(2)) ⊂ F h(C(2)). To conclude the equality F h(A(2)) = F h(C(2)) we have to prove for x ∈ F h(A(2)) and x ∈ (F h(C(2)))2 that x ∈ (F h(A(2)))2. In other words there is no quadratic extension between F h(A(2)) and F h(C(2)). Take x which satis- fies these conditions. From Remark 2.9 we can assume, without loss of generality, that x ∈ F×. Then x ∈ (F h(C(2)))2 ∩ F× = (1 +mC)(F ×)2. Since by Corollary 2.5, (1+mA)(F ×)2 = R(F ) = (1+mC)(F ×)2 it follows that x ∈ (1 +mA)(F ×)2 ⊂ (F h(A(2)))2, proving the desired equality. We now prove the uniqueness of such an extension C(2). According to the Conjugation Theorem [7, 3.2.15] for two extensions C(2) and C(2)′ of C to F (2) there exists g ∈ GF (2) such that C(2)′ = g(C(2)). If C(2) and C(2)′ are finer than A(2), then g(A(2)) and A(2) are both coarser than C(2)′, which implies that g(A(2)) and A(2) have to be comparable by inclusion. Thus g(A(2)) = A(2), by [7, Lemma 3.2.8]. Hence g ∈ Gh(A(2)). Since we know Gh(A(2)) = Gh(C(2)), g ∈ Gh(C(2)) and C(2)′ = C(2), as claimed. 46 2-Galois groups and the Kaplansky radical Consider next the case where A and C are not comparable. Then A, C ∈ Ω2, and we know by Proposition 2.6, (3) and (5), that D = A ·C ∈ Ω2 and ΓD 6= 2ΓD. Since A and D, respectively C and D, are comparable by inclusion, by the first part of the proof there is an extension D(2) of D to F (2) which contains uniquely determined extensions A(2) and C(2), respectively, of A and C to F (2) for which Gh(A(2)) = Gh(D(2)) and Gh(C(2)) = Gh(D(2)). Thus Gh(A(2)) = Gh(C(2)). In the next result we complete the study of pre-2-henselian valued fields (F,A) by considering the missing case ΓA = 2ΓA in Theorem 4.2. Recall that Ω1 is totally ordered by inclusion, Proposition 2.6 (2). Hence A(1) = ⋂ A∈Ω1 A is a valuation ring having maximal ideal m(1) = ⋃ A∈Ω1 mA. Thus A(1) is a pre-2-henselian valuation ring of F . Observe also that A(1) is comparable by inclusion to any other element of Ω, Proposition 2.6 (4). Theorem 4.7. Let F be a field with R(F ) 6= F× and Ω 6= {F }, but each C ∈ Ω has 2-divisible value group. Let k be the residue class field of the valuation ring A(1) introduced above. Then GF (2) = F ∗Gk(2), and the following conditions hold: (1)F is a free pro-2 group and rank(F) = rank ( (1 +m(1))(F ×)2/(F×)2 ) (2) k does not admit any proper pre-2-henselian valuation ring. Moreover, k×/(k×)2 ∼= F×/(1 +m(1))(F ×)2. Conversely, if there exists a decomposition GF (2) = F ∗GH(2) where F is a free pro-2 group and H has a proper 2-henselian valuation ring C with ΓC = 2ΓC and residue class field k which does not admit any proper pre-2-henselian valuation ring, then R(F ) 6= F×, Ω 6= {F } and for every A ∈ Ω, ΓA = 2ΓA. Proof. Since Ω 6= {F }, A(1) is a proper valuation ring of F . By applying Proposition 4.4 to A(1) we get GF (2) = GK(2)∗GFh (2), where F = GK(2) is free and Fh is a 2-henselization of (F,A(1)). Thus (F,A(1)) and (Fh, A) have the same value group and the same residue class field. By assumption A(1), and consequently also A, have 2-divisible value group. Let A(2) be the extension of A to F (2). Since GFh (2) is a pro-2 group and A has 2-divisible value group, we can conclude that A(2) has a trivial inertia group over Fh. Thus GFh (2) ∼= Gk(2), where k is the residue class field of A and also of A(1) (see [5, Theorem 19.6]). To complete the first part of the R. P. Dario, A. J. Engler 47 proof it remains to show that k does not admit any proper pre-2-henselian valuation ring. According to Proposition 2.6 (5), Ω2 = ∅. Hence Ω = Ω1 and therefore A(1) ∈ Ω1 does not contain any proper pre-2-henselian valuation ring of F . Denote by π the canonical map corresponding to A(1) and k and recall from Proposition 2.4 (2) that π(R(F )) = R(k). Since π induces a bijection between the set of all valuation rings of k and the set of all valuation rings of F finer than A(1) ([5, Theorem 8.7]), we can conclude that k does not admit any proper pre-2-henselian valuation ring, as claimed in (2). The isomorphism k×/(k×)2 ∼= F×/(1 +m(1))(F ×)2 follows from F× = A× (1)(F ×)2, because ΓC is 2-divisible. Conversely, given a decomposition GF (2) = F ∗GH(2) we can apply Proposition 4.5 to H and to the fixed field K of F . Then A = C ∩ F is a proper pre-2-henselian valuation ring of F and we may assume that (H,C) is a 2-henselization of (F,A). Since ΓC is a 2-divisible group and (H,C) is an immediate extension of (F,A), also ΓA = 2ΓA. Moreover kA = kC does not admit any pre-2-henselian valuation ring. In particular kA is not quadratically closed and so A ∈ Ω1. Moreover, the bijective correspondence between the set of valuation rings of kA and the set of valuation rings of F finer than A implies that A = A(1) and Ω2 = ∅. Hence any element of Ω, being coarser than A(1) which has a 2-divisible value group, also has a 2-divisible value group. Finally, R(F ) 6= F× follows from the properties of A combined with (1) of Proposition 2.4. 5. Examples and comments The simplest possible description of GF (2) for a pre-2-henselian valued field (F,A) occurs for A ∈ Ω2. Then GF (2) = F ∗ Zµ 2 where µ = rank(F×/R(F )). Indeed, observe that R(F ) = (1 +mA)(F ×)2 (Proposi- tion 2.6 (5)) and A× ⊂ (1+mA)(F ×)2 (A ∈ Ω2). Thus A×(F×)2 = R(F ). Finally ΓA/2ΓA ∼= F×/A×(F×)2. Theorems 4.2 and 4.7 can be seen as tools to produce fields with Kaplansky radical having some prescribed conditions. By [9, Theorem 3.6] given two pro-2 groups F and H which can be realized as Galois groups GK(2) and GH(2), for some fields K and H, there exists a field F such that GF (2) ∼= F ∗ H. Thus, all we need to construct fields with a pre-2- henselian valuation ring is to choose fields K and H having convenient properties. Examples of fields K for which GK(2) is a free pro-2 group are well-known. Let us fix that K has characteristic 6= 2 and GK(2) is a free pro-2 group. To get suitable examples of 2-henselian valued fields it is enough to take H = k((X))Γ, the field of generalized formal power series 48 2-Galois groups and the Kaplansky radical over a field k of characteristic 6= 2 and with respect to a totally ordered abelian group Γ. H has a henselian valuation ring AH = k[[X]]Γ which is a fortiori 2-henselian. Moreover k and Γ are, respectively, residue class field and value group of AH . Therefore if we choose k a pre-Hilbertian (as in 1.4) field with an Archimedean ordering and Γ a 2-divisible group, then we get a field F for which GF (2) = GK(2) ∗ GH(2) exemplifies Theorem 4.7. Here A(1) = AH ∩ F . Similarly, the choice of k quadratically closed and Γ 6= 2Γ will give us examples of pre-2-henselian valued fields F with Ω2 6= ∅. In this case GF (2) = F ∗ Zµ 2 , as described in the first paragraph. Adding to K the property −1 6∈ (K×)2 while −1 ∈ (k×)2 we get a pre- 2-henselian valued field F with −1 ∈ R(F )r(F×)2. Hence F× = DF 〈1, 1〉 and so F has level s = 2 (the least integer n ≥ 1 such that −1 can be expressed as a sum of n squares in F ). Now, if we impose Γ 6= 2Γ it follows that F has R(F )-rigid elements. A fact that cannot happen with the usual rigid elements: by [4, Lemma 3.6], if every x ∈ F× is a sum of s(F ) squares in F , then no element of F is (F×)2-rigid. Therefore we cannot just translate to R(F )-rigid elements the properties of rigid elements. More generally, choose K and H as above and also satisfying rank (K×/(K×)2) = m and rank (H×/(H×)2) = n, for some integers m > 0 and n > 1 and Γ 6= 2Γ. Then we get a pre-2-henselian valued field F such that rank (R(F )/(F×)2) = m and rank (F×/R(F )) = n. The last statement is nearly the content of [2, Corollary 3.13]. The field presented in [2, Theorem 2.3] satisfies −1 ∈ (F×)2 and (F× : R(F )) = 4. An easy calculation shows that B(R(F )) = R(F ) and consequently there is a pre-2-henselian valuation ring A such that ΓA 6= 2ΓA. The same is true for fields constructed using the recipe of [2, Theorem 3.12]. Some other examples of fields with non-trivial Kaplansky radical, i.e., R(F ) 6= (F×)2, F×, like the one due to Gross-Fischer [8, Examples, p 306-307] or the examples [13, (A), (C), and (D)] constructed by Kula, are all pre-2-henselian valued fields. Therefore, our method described above gives a large family of examples, many of them have been construct before using different particular methods. For the next application let us recall that XF usually stands for the space of orderings of a field F and it has naturally the structure of a Boolean topological space (see for example [14, Chapter VIII, §6]). If F is formally real, then R(F ) ⊂ DF 〈1, 1〉 ⊂ ∑ (F×)2 and so a pre-2-henselian valuation ring A of F is compatible with all orderings of F . This is a property which (F,A) shares with any of its henselizations (Fh, Ah), as one deduces from [15, Theorem 3.16]. Moreover, by applying [15, Theorem 3.10] to (F,A) and (Fh, Ah) we get that the map P 7→ P ∩ F from XFh to R. P. Dario, A. J. Engler 49 XF is bijective. Since this map is always continuous, compactness implies that the map is an homeomorphism. The above comments show that we can also prescribe the space of orderings XF in constructing examples of pre-2-henselian valued fields. It suffices to choose H having the desired XH . A final word, theorems 4.2 and 4.7 have a corresponding formulation in Witt ring language. References [1] J. K. Arason, R. Elman, B. Jacob, Rigid elements, valuations, and realization of Witt rings, J. Algebra, N.110, 1987, pp. 449-467. [2] L. Berman, Pythagorean fields and the Kaplansky radical, J. Algebra N.61, 1979, pp. 497-507. [3] C.M. Cordes, Kaplansky’s radical and quadratic forms over non-real fields, Acta Arith., N.28, 1975, pp. 253-271. [4] R. Elman, T.Y. Lam, Quadratic forms and the u-invariant I, Math. Z., N.131, 1973, pp. 282-304. [5] O. Endler, Valuation Theory, Springer, Berlin, 1972. [6] A. J. Engler, J. Koenigsmann, Abelian subgroups of pro-p Galois groups, Trans. Amer. Math. Soc., N. 350, 1996, pp. 2473-2485. [7] A. J. Engler, A. Prestel, Valued Fields, Springer, Berlin, 2005. [8] H. Gross, H. R. Fischer, Non real fields k and infinite dimensional k-vectorspaces (Quadratic forms and linear topologies, II), Math. Ann. N., 159, 1965, pp. 285-308. [9] B. Jacob, R. Ware, A recursive description of the maximal pro-2 Galois group via Witt rings, Math. Z., N.200, 1989, pp. 379-396. [10] I. Kaplansky, Frцhlich’s local quadratic forms, J. Reine Angew. Math., N.239, 1969, pp. 74-77. [11] M. Knebusch, On the extension of real places, Comment. Math. Helv., N.48, 1973, pp. 354-369. [12] J. Koenigsmann, Encoding valuations in absolute Galois groups, Valuation Theory and Its Applications, Vol. II (Saskatoon, SK, 1999), Fields Inst. Commun., Vol. 33, American Mathematical Society, Rhode Island, 2003, pp. 107-132. [13] M. Kula, Fields with non-trivial Kaplansky’s radical and finite square class number. Acta Arith., N. 38, 1981, pp. 411-418. [14] T. Y. Lam, Introduction to Quadratic Forms over Fields, AMS, Rhode Island, 2004. [15] T. Y. Lam, Orderings, Valuations and Quadratic Forms, Conference Board of the Mathematical Science, N. 52. Providence, RI: Amer. Math. Soc. 1983. [16] A.S. Merkurjev, On the norm residue symbol of degree 2, Dokl. Akad. Nauk., N.261, 1981, pp. 542-547; English transl. Dokl. Math. N.24, 1981, pp. 546-551. [17] J. Neukirch, Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Arch. Math.(Basel), N.22, 1971, pp. 337-357. [18] L. Ribes, Introduction of Profinite Groups and Galois Cohomology, Queen’s Papers in Pure and Applied Math., N.24, Queen’s University, Kingston, Ontario, 1970. [19] R. Ware, Valuations rings and rigid elements in fields, Canad. J. Math. 33, 1981, pp. 1338-1355. 50 2-Galois groups and the Kaplansky radical Contact information R. P. Dario UTFPR-DAMAT, Av. Sete de Setembro, 3165, 80230-901 Curitiba, PR, Brasil E-Mail: ronie@utfpr.edu.br URL: http://pessoal.utfpr.edu.br/ronie A. J. Engler UNICAMP-IMECC, Caixa Postal 6065, 13083-970 Campinas, SP, Brasil E-Mail: engler@ime.unicamp.br URL: www.ime.unicamp.br/∼engler Received by the editors: 31.10.2009 and in final form 01.03.2011.

2-Galois groups and the Kaplansky radical

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