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CONSTRUCTION OF UNIPOTENT GALOIS EXTENSIONS AND MASSEY PRODUCTS JÁNMINÁCˇ ANDNGUYỄNDUYTÂN 6 1 DedicatedtoAlexanderMerkurjev 0 2 p ABSTRACT. For all primes p and for all fields, we find a sufficient and necessary con- e dition of the existence of a unipotent Galois extension of degree p6. The main goal of S this paper is to describe an explicit construction of such a Galois extension over fields 6 admitting such a Galois extension. This construction is surprising in its simplicity and 1 generality. The problem of finding such a construction has been left open since 2003. Recently a possible solution of this problem gained urgency because of an effortto ex- ] T tend new advances in Galois theory and its relations with Massey products in Galois N cohomology. . h t a m 1. INTRODUCTION [ FromtheverybeginningoftheinventionofGaloistheory, oneproblemhasemerged. 4 v For a given finite group G, find a Galois extension K/Q such that Gal(K/Q) G. This 6 ≃ is still an open problem in spite of the great efforts of a number of mathematicians and 4 3 substantialprogresshavingbeenmadewithspecificgroups G. (See[Se3].) Amoregen- 1 eral problem is to ask the same question over other base fields F. This is a challenging 0 and difficult problem even for groups Gof primepower order. . 1 In this paper we make progress on this classical problem in Galois theory. More- 0 5 over this progress fits together well with a new development relating Massey prod- 1 ucts in Galois cohomology to basic problems in Galois theory. For all primes p and all : v fields in the key test case of n = 4, we construct Galois extensions with the unipotent i X Galois group U (F ) assuming only the existence of some Galois extensions of order n p r p3. This fits into a program outlined in [MT1] and [MT2], for the systematic construc- a tion ofGalois p-closed extensionsofgeneralfields,assumingonlyknowledge ofGalois extensions of degree less than or equal to p3 and the structure of pth power roots of unity in the base field. Thus both the methods and the results in this paper pave the way to a program for obtaining the structure of maximal pro-p-quotients of absolute Galois groups for all fields. We shall now describe some previous work of a number of mathematicians which has influenced our work, as well as its significance for further developmentsand applications. JM is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC)grantR0370A01. NDTispartiallysupportedbytheNationalFoundationforScienceandTech- nologyDevelopment(NAFOSTED)grant101.04-2014.34. 1 2 JÁNMINÁCˇ ANDNGUYỄNDUYTÂN RecentlytherehasbeensubstantialprogressinGaloiscohomologywhichhaschanged our perspective on Galois p-extensions over general fields. In some remarkable work, M. Rost and V. Voevodsky proved the Bloch-Kato conjecture on the structure of Galois cohomology of general fields. (See [Voe1, Voe2].) From this work it follows that there must be enough Galois extensions to make higher degree Galois cohomology decom- posable. However the explicit construction of such Galois extensions is completely mysterious. In [MT1], [MT2] and [MT5], two new conjectures, the Vanishing n-Massey Conjecture and the Kernel n-Unipotent Conjecture were proposed. These conjectures in [MT1] and [MT2], and the results in this paper, lead to a program of constructing these previously mysterious Galois extensions in a systematic way. In these papers it is shown that the truth of these conjectures has some significant implications on the structure of absolute Galois groups. These conjectures are based on a number of previous considerations. One motivation comes from topological considerations. (See [DGMS] and [HW].) Another motivation is a program to describe various n-central series of absolute Galois groups as kernels of simple Galois representations. (See [CEM, Ef, EM1, EM2, MSp, NQD, Vi].) If the Vanishing n-Massey Conjecture is true, then by a result in [Dwy], we obtain a program of building up n-unipotent Galois representations of absolute Galois groups by induction on n. This is an attractive program because we obtain a procedure of constructing larger Galois p-extensions from smaller ones, efficiently using the fact that certain a priori natural cohomological obstructions tothis procedure alwaysvanish. Recallthatforeachnaturalnumbern,U (F )isthegroupofuppertriangularn n- n p × matrices with entries in F and diagonal entries 1. Then U (F ) is isomorphic to the p 3 2 dihedral group of order 8, and if p is odd, then U (F ) is isomorphic to the Heisen- 3 p berg group H of order p3. For all n 4 and all primes p, we can think of U (F ) as p3 ≥ n p "higher Heisenberg groups" of order pn(n 1)/2. It is now recognized that these groups − playaveryspecialroleincurrentGaloistheory. BecauseU (F )isaSylow p-subgroup n p of GL (F ), andevery finite p-group hasa faithful linear n-dimensional representation n p over F , for some n, we see that every finite p-group can be embedded into U (F ) p n p for some n. Besides, the Vanishing n-Massey Conjecture and the Kernel n-Unipotent Conjecture also indicate some deeper reasons why U (F ) is of special interest. The n p constructions of Galois extensions with the Galois group U (F ) over fields which ad- 3 p mitthem,arewell-knowninthecasewhenthebasefieldisofcharacteristic not p. They are an important basic tool in the Galois theory of p-extensions. (See for example [JLY, Sections6.5and6.6]. Someearlypapersrelatedtothesetopicslike[MNg]and[M]now belongto classical work on Galoistheory.) In [GLMS, Section 4], a construction of Galois extensions K/F, char(F) = 2, with 6 Gal(K/F) U (F ), was discovered. Already at that time, one reason for search- 4 2 ≃ ing for this construction was the motivation to find ideas to extend deep results on the characterization of the fixed field of the third 2-Zassenhaus filtration of an absolute Galois group G as the compositum of Galois extensions of degree at most 8 (see F CONSTRUCTIONOFUNIPOTENTGALOISEXTENSIONSAND MASSEYPRODUCTS 3 [Ef, EM2, MSp, Vi]), to a similar characterization of the fixed field of the fourth 2- Zassenhaus filtration of G . In retrospect, looking at this construction, one recognizes F some elements of the basic theory of Massey products. However at that time the authors of [GLMS] were not familiar with Massey products. It was realized that such a construction would also be desirable for U (F ) for all p rather than U (F ), but none 4 p 4 2 hasbeenfound until now. In [GLMS], in the construction of a Galois field extension K/F with Gal(K/F) ≃ U (F ), a simple criteria was used for an element in F to be a norm from a bicyclic 4 2 extension of degree 4 modulo non-zero squares in the base field F. However in [Me], A. Merkurjev showed that a straightforward generalization of this criteria for p odd instead of p = 2, is not true in general. Therefore it was not clear whether such an analogous construction of Galois extensions K/F with Gal(K/F) U (F ) was possi- 4 p ≃ ble for p odd. On the other hand, a new consideration in [HW], [MT1] and [MT2] led us to formu- late the Vanishing n-Massey Conjecture, andthe most natural wayto prove thisconjec- ture for n = 3 in the key non-degenerate case would be through constructing explicit Galois U (F )-extensions. In fact we pursued both cohomological variants of prov- 4 p ing the Vanishing 3-Massey Conjecture and the Galois theoretic construction of Galois U (F )-extensions. 4 p ThestoryofprovingthisconjectureandfinallyconstructingGaloisU (F )-extensions 4 p overallfieldswhichadmitthem,isinteresting. FirstM.J.HopkinsandK.G.Wickelgren in[HW]provedaresultwhich impliesthattheVanishing3-MasseyConjecture with respect to prime 2, is true for all global fields of characteristic not 2. In [MT1] we proved that the result of [HW] is valid for any field F. At the same time, in [MT1] the Van- ishing n-Massey Conjecture was formulated, and applications on the structure of the quotients of absolute Galois groups were deduced. In [MT3] we proved that the Van- ishing 3-Massey Conjecture with respect to any prime p is true for any global field F containing a primitive p-th root of unity. In [EMa1], I. Efrat and E. Matzri provided alternative proofs for the above-mentioned results in [MT1] and [MT3]. In [Ma], E. Matzri proved that for any prime p and for any field F containing a primitive p-th root ofunity,everydefinedtripleMasseyproductcontains0. ThisestablishedtheVanishing 3-Massey Conjecture in the form formulated in [MT1]. Shortly after [Ma] appeared on the arXiv, two new preprints, [EMa2] and [MT5], appeared nearly simultaneously and independently on the arXiv as well. In [EMa2], I. Efrat and E. Matzri replace [Ma] and provide a cohomological approach to the proof of the main result in [Ma]. In [MT5] we also provide a cohomological method of proving the same result. We also extend the vanishing of triple Massey products to all fields, and thus remove the restriction that thebasefieldcontainsaprimitive p-throotofunity. Wefurtherprovideapplicationson thestructure ofsomecanonical quotientsofabsolute Galoisgroups, andalsoshowthat some specialhigher n-fold Masseyproducts vanish. Finallyinthispaperwe areableto provide a construction of the Galois U (F )-extension M/F for any field F which ad- 4 p mitssuch an extension. Weuse thisconstruction toprovide anatural newproof, which 4 JÁNMINÁCˇ ANDNGUYỄNDUYTÂN we were seeking from the beginning of our search for a Galois theoretic proof, of the vanishing oftriple Massey products overall fields. Someinteresting casesof"automatic" realizationsofGaloisgroupsare known. These are cases when the existence of one Galois group over a given field forces the existence of some other Galois groups over this field. (See for example [Je, MS2, MSS, MZ, Wh].) However,nontrivialcasesofautomaticrealizationscomingfrom anactualconstruction of embedding smaller Galois extensions to larger ones, are relatively rare, and they are difficult to produce. In our construction we are able, from knowledge of the existence of two Heisenberg Galois extensions of degree p3 over a given base field F as above, to find possibly another pair of Heisenberg Galois extensions whose compositum can be automatically embeddedin a Galois U (F )-extension. (See also Remark3.3.) Observe 4 p thatin all proofs ofthe Vanishing3-Massey Conjecture we currently have, constructing HeisenbergGaloisextensionsofdegree p3 hasplayedanimportantrole. Forthesakeof a possible inductive proof of the Vanishing n-Massey Conjecture, it seemsimportant to beabletoinductivelyconstruct GaloisU (F )-extensions. Thishasnowbeenachieved n p for the induction step from n = 3 to n = 4, and it opens up a way to approach the Vanishing4-Massey Conjecture. Anothermotivationforthisworkwhichcombineswellwiththemotivationdescribed above, comes from anabelian birational considerations. Very roughly in various generality and precision, it was observed that small canonical quotients of absolute Galois groups determine surprisingly precise information about base fields, in some cases en- tire base fields up to isomorphisms. (See [BT1, BT2, CEM, EM1, EM2, MSp, Pop].) But these results suggest that some small canonical quotients of an absolute Galois group together with knowledge of the roots of unity in the base field should determine larger canonical quotients of this absolute Galois group. The Vanishing n-Massey Conjecture andthe Kernel n-Unipotent Conjecture, together with the program ofexplicit constructions of Galois U (F )-extensions, make this project more precise. Thus our main re- n p sults, Theorems3.7, 3.9, 4.3 and5.5,contribute to this project. A further potentially important application for this work is the theory of Galois p- extensions of global fields with restricted ramification and questions surrounding the Fontaine-Mazur conjecture. (See [Ko], [La], [McL], [Ga],[Se2].) For example in [McL, Section 3], there is a criterion for infinite Hilbert p-class field towers over quadratic imaginary number fields relying on the vanishing of certain triple Massey products. Theexplicitconstructions inthispapershould beusefulforapproachingtheseclassical numbertheoretic problems. Only relatively recently, the investigations of the Galois realizability of some larger p-groups among families of small p-groups, appeared. (See the very interesting papers [Mi1], [Mi2], [GS].) In these papers the main concern is understanding cohomological and Brauer group obstructions for the realizability of Galois field extensions with pre- scribed Galois groups. In our paper the main concern is the explicit constructions and their connections with Massey products. In other recent papers [CMS] and [Sch], the CONSTRUCTIONOFUNIPOTENTGALOISEXTENSIONSAND MASSEYPRODUCTS 5 authors succeededto treat the casesofcharacteristic equal to p ornot equalto p, nearly uniformly. Thisisalso the case with our paper. Our paper is organized as follows. In Section 2 we recall basic notions about norm residue symbols and Heisenberg extensions of degree p3. (For convenience we think of the dihedral group of order 8 as the Heisenberg group of order 8.) In Section 3 we provide adetailed construction ofGaloisU (F )-extensions beginningwith two "com- 4 p patible" Heisenberg extensions of degree p3. Section 3 is divided into two subsections. In Subsection 3.1 we provide a construction of the required Galois extension M/F over any field F which contains a primitive p-th root of unity. In Subsection 3.2 we provide suchaconstructionforallfieldsofcharacteristicnot p,buildingontheresultsandmeth- odsinSubsection 3.1. InExample3.8weillustrate ourmethodonasurprisinglysimple construction of Galois U (F )-extensions over any field F with char(F) = 2. In Section 4 2 6 4 we provide a required construction for all fields of characteristic p. After the original andclassical papersofE. Artin andO. Schreier[ASch]and E.Witt[Wi], these constructions seem to add new results on the construction of basic Galois extensions M/F with Galois groups U (F ), n = 3 and n = 4. These are aesthetically pleasing constructions n p with remarkable simplicity. They follow constructions in characteristic not p, but they aresimpler. Seealso[JLY,Section5.6andAppendixA1]foranotherproceduretoobtain theseGaloisextensions. InSection 5weprovide anewnatural Galoistheoretic proofof the vanishing of triple Massey products over all fields in the key non-degenerate case. We also complete the new proof of the vanishing of triple Massey products in the case whenaprimitive p-throotofunityiscontainedinthebasefield. Finallyweformulatea necessary and sufficient condition for the existence of a Galois U (F )-extension M/F 4 p which contains an elementary p-extension of any field F (described by three linearly independentcharacters), and we summarizethe mainresults in Theorem 5.5. Acknowledgements: Wewould liketothankM.Ataei,L.Bary-Soroker, S. K.Chebolu, I. Efrat, H. Ésnault, E. Frenkel, S. Gille, J. Gärtner, P. Guillot, D. Harbater, M. J. Hop- kins, Ch. Kapulkin, I. Krˇíž, J. Labute, T.-Y. Lam, Ch. Maire, E. Matzri, C. McLeman, D. Neftin, J. Nekovárˇ, R. Parimala, C. Quadrelli, M. Rogelstad, A. Schultz, R. Sujatha, Ng. Q. Thắng, A. Topaz, K. G. Wickelgren and O. Wittenberg for having been able to share our enthusiasm for this relatively new subject of Massey products in Galois cohomology, and for their encouragement, support, and inspiring discussions. We are very grateful to the anonymous referee for his/her careful reading of our paper, and for providing us with insightful comments and valuable suggestions which we used to improve ourexposition. Notation: IfGisagroupandx,y G,then[x,y]denotesthecommutatorxyx 1y 1. For − − ∈ anyelement σ of finite order n in G, we denote N to be the element1+σ+ +σn 1 σ − ··· in the integral group ringZ[G] ofG. ForafieldF,wedenoteF (respectivelyG )tobeitsseparableclosure(respectivelyits s F absolute Galois group Gal(F /F)). We denote F to be the set of non-zero elements of s × 6 JÁNMINÁCˇ ANDNGUYỄNDUYTÂN F. For a given profinite group G, we call a Galois extension E/F, a (Galois) G-extension ifthe Galois group Gal(E/F) isisomorphic to G. Foraunitalcommutative ring Randanintegern 2,wedenoteU (R) asthegroup n ≥ ofallupper-triangularunipotentn n-matriceswithentriesin R. Forany(continuous) × representation ρ: G U (R) from a (profinite) group G to U (R) (equipped with n n → discrete topology ), and 1 i < j n, let ρ : G R be the composition of ρ with the ij ≤ ≤ → projection from U (R) to its (i,j)-coordinate. n 2. HEISENBERG EXTENSIONS Thematerials inthis section have beentaken from [MT5, Section 3]. 2.1. Normresiduesymbols. Let F beafieldcontainingaprimitive p-throotofunityξ. For any element a in F , we shall write χ for the character corresponding to a via the × a Kummermap F H1(G ,Z/pZ) = Hom(G ,Z/pZ). Fromnowonweassumethat × F F → a is not in (F )p. The extension F(√p a)/F is a Galois extension with the Galois group × σ Z/pZ, where σ satisfies σ (√p a) = ξ√p a. a a a h i ≃ Thecharacterχ definesahomomorphism χa Hom(G , 1Z/Z) Hom(G ,Q/Z) a ∈ F p ⊆ F bythe formula 1 χa = χ . a p Let b be anyelementin F . Then the norm residue symbol maybe definedas × (a,b) := (χa,b) := b δχa. ∪ Here δ is the coboundary homomorphism δ: H1(G,Q/Z) H2(G,Z) associated to → the short exact sequenceof trivial G-modules 0 Z Q Q/Z 0. → → → → The cup product χ χ H2(G ,Z/pZ) can be interpreted as the norm residue a b F ∪ ∈ symbol (a,b). More precisely, we considerthe exact sequence x xp 0 −→ Z/pZ −→ Fs× −7→→ Fs× −→ 1, where Z/pZ hasbeen identified with the group of p-th roots of unity µ via the choice p of ξ. As H1(G ,F ) = 0, weobtain F s× 0 H2(G ,Z/pZ) i H2(G ,F ) ×p H2(G ,F ). −→ F −→ F s× −→ F s× Then one hasi(χ χ ) = (a,b) H2(G ,F ). (See[Se1, ChapterXIV, Proposition 5].) a ∪ b ∈ F s× CONSTRUCTIONOFUNIPOTENTGALOISEXTENSIONSAND MASSEYPRODUCTS 7 2.2. Heisenbergextensions. InthissubsectionwerecallsomebasicfactsaboutHeisen- berg extensions. (See[Sha, Chapter2, Section 2.4]and[JLY,Sections 6.5and 6.6].) Assume that a,b are elements in F , which are linearly independent modulo (F )p. × × Let K = F(√p a,√p b). Then K/F is a Galois extension whose Galois group is generated by σ and σ . Here σ (√p b) = √p b, σ (√p a) = ξ√p a; σ (√p a) = √p a, σ (√p b) = ξ√p b. a b a a b b 1 x z We consider a map U (Z/pZ) (Z/pZ)2 which sends 0 1 y to (x,y). Then 3 →   0 0 1 we havethe following embeddingproblem   G F ρ¯ (cid:15)(cid:15) 0 // Z/pZ // U (Z/pZ) // (Z/pZ)2 // 1, 3 where ρ¯ is the map (χ ,χ ): G Gal(K/F) (Z/pZ)2. (The last isomorphism a b F → ≃ Gal(K/F) (Z/pZ)2 isthe one which sends σ to (1,0) and σ to (0,1).) a b ≃ Assume that χ χ = 0. Then the norm residue symbol (a,b) is trivial. Hence there a b ∪ exists α in F(√p a) such that N (α) = b (see[Se1, ChapterXIV,Proposition 4(iii)]). F(√p a)/F Weset p 2 A0 = αp−1σa(αp−2)···σap−2(α) = ∏− σai(αp−i−1) ∈ F(√p a). i=0 Lemma2.1. Let f bean elementin F . Let A = f A . Thenwe have a × a 0 σa(A) NF(√p a)/F(α) b = = . A αp αp σ (A) σ (A ) a a 0 Proof. Observe that = . The lemmathen follows from the identity A A 0 p 2 p 1 (s 1) ∑− (p i 1)si = ∑− si ps0. (cid:3) − − − − i=0 i=0 Proposition 2.2. Assume that χ χ = 0. Let f be an element in F . Let A = f A be a b a × a 0 ∪ defined as above. Then the homomorphism ρ¯ := (χ ,χ ): G Z/pZ Z/pZ lifts to a a b F → × Heisenbergextension ρ: G U (Z/pZ). F 3 → Sketch of Proof. Let L := K(√p A)/F. Then L/F is Galois extension. Let σ˜ Gal(L/F) a ∈ (resp. σ˜ Gal(L/F)) be an extension of σ (resp. σ ). Since σ (A) = A, we have b a b b ∈ σ˜ (√p A) = ξj√p A, for some j Z. Hence σ˜p(√p A) = √p A. This implies that σ˜ is of b ∈ b b order p. b √p b On the other hand, we have σ˜ (√p A)p = σ (A) = A . Hence σ˜ (√p A) = ξi√p A , a a αp a α for some i Z. Then σ˜p(√p A) = √p A. Thus σ˜ isof order p. a a ∈ 8 JÁNMINÁCˇ ANDNGUYỄNDUYTÂN If we set σ := [σ˜ ,σ˜ ], then σ (√p A) = ξ 1√p A. This implies that σ is of order p. A a b A − A Alsoone can checkthat [σ˜ ,σ ] = [σ˜ ,σ ] = 1. a A b A Wecan definean isomorphism ϕ: Gal(L/F) U (Z/pZ) byletting 3 → 1 1 0 1 0 0 1 0 1 σ 0 1 0 ,σ 0 1 1 ,σ 0 1 0 . a b A 7→   7→   7→   0 0 1 0 0 1 0 0 1       ϕ Then the composition ρ: G Gal(L/F) U (Z/pZ) isthe desired lifting of ρ¯. F 3 → → Note that [L : F] = p3. Hence there are exactly p extensions of σ Gal(E/F) to the a ∈ automorphisms in Gal(L/F) since [L : E] = p3/p2 = p. Therefore for later use, we can choose an extension, still denoted by σ Gal(L/F), of σ Gal(K/F) in such a way a a ∈ ∈ √p b that σ (√p A) = √p A . (cid:3) a α 3. THE CONSTRUCTION OF U4(Fp)-EXTENSIONS: THE CASE OF CHARACTERISTIC = p 6 3.1. Fieldscontainingprimitive p-throotsofunity. Inthissubsection weassumethat F is a field containing a primitive p-th root ξ of unity. The following result can be deduced from Theorem 5.5, but for the convenience of the reader we include a proof here. Proposition 3.1. Assume that there exists a Galois extension M/F such that Gal(M/F) ≃ U (F ). Then there exist a,b,c F such that a,b,c are linearly independent modulo (F )p 4 p × × ∈ and (a,b) = (b,c) = 0. Moreover M contains F(√p a,√p b,√p c). Proof. Let ρ be the composite ρ: G ։ Gal(M/F) U (F ). Then ρ ,ρ and ρ are F 4 p 12 23 34 ≃ elements in Hom(G ,F ). Hence there are a,b and c in F such that χ = ρ , χ = ρ F p × a 12 b 23 and χ = ρ . Since ρ is a group homomorphism, by looking at the coboundaries of ρ c 34 13 and ρ , wesee that 24 χ χ = χ χ = 0 H2(G ,F ). a b b c F p ∪ ∪ ∈ Thisimpliesthat (a,b) = (b,c) = 0 by[Se1, ChapterXIV, Proposition 5]. Let ϕ := (χ ,χ ,χ ): G (F )3. Then ϕ is surjective. By Galois correspondence, a b c F p → we have Gal(F /F(√p a,√p b,√p c)) = kerχ kerχ kerχ = kerϕ. s a b c ∩ ∩ This implies that Gal(F(√p a,√p b,√p c)/F) (F )3. Hence by Kummer theory, we see p ≃ thata,bandcarelinearlyindependentmodulo(F )p. Clearly, McontainsF(√p a,√p b,√p c). × (cid:3) Conversely we shall see in this section that given these necessary conditions for the existence of U (F )-Galois extensions over F, as in Proposition 3.1, we can construct a 4 p Galoisextension M/F with the Galois group isomorphic to U (F ). 4 p CONSTRUCTIONOFUNIPOTENTGALOISEXTENSIONSAND MASSEYPRODUCTS 9 Fromnowonweassumethatwearegivenelementsa,bandcinF suchthata,band × carelinearlyindependentmodulo(F )p andthat(a,b) = (b,c) = 0. Weshallconstruct × a GaloisU (F )-extension M/F such that M contains F(√p a,√p b,√p c). 4 p FirstwenotethatF(√p a,√p b,√p c)/F isaGaloisextensionwithGal(F(√p a,√p b,√p c)/F) generated by σ ,σ ,σ . Here a b c σ (√p a) = ξ√p a,σ (√p b) = √p b,σ (√p c) = √p c; a a a σ (√p a) = √p a,σ (√p b) = ξ√p b,σ (√p c) = √p c; b b b σ (√p a) = √p a,σ (√p b) = √p b,σ (√p c) = ξ√p c. c c c Let E = F(√p a,√p c). Since (a,b) = (b,c) = 0, there are α in F(√p a) and γ in F(√p c) (see [Se1, ChapterXIV, Proposition 4(iii)]) such that N (α) = b = N (γ). F(√p a)/F F(√p c)/F Let G be the Galois group Gal(E/F). Then G = σ ,σ , where σ G (respec- a c a h i ∈ tively σ G) is the restriction of σ Gal(F(√p a,√p b,√p c)/F) (respectively σ c a c ∈ ∈ ∈ Gal(F(√p a,√p b,√p c)/F)). Our next goal is to find an element δ in E such that the Galois closure of E(√p δ) is × our desired U (F )-extension of F. Wedefine 4 p p 2 C = ∏− σi(γp i 1) F(√p a), 0 c − − ∈ i=0 anddefineB := γ/α. Thenwehavethefollowingresult,whichfollowsfromLemma2.1 (see [Ma, Proposition 3.2]and/or [MT5, Lemma4.2]). Lemma3.2. We have σ (A ) a 0 (1) = N (B). A σc 0 σ (C ) (2) c 0 = N (B) 1. (cid:3) C σa − 0 Remark3.3. Wewouldliketoinformallyexplainthemeaningofthenextlemma. From our hypothesis (a,b) = 0 = (b,c) and from Subsection 2.2, we see that we can obtain twoHeisenbergextensions L = F(√p a,√p b,√p A ) and L = F(√p b,√p c,√p C ) of F. Here 1 0 2 0 we have chosen specific elements A F(√p a) and C F(√p c). However we may not 0 0 ∈ ∈ be able to embed the compositum of L and L into our desired Galois extension M/F 1 2 withGal(M/F) U (F ). WeknowthatwecanmodifytheelementA byanyelement 4 p 0 ≃ f F and the element C by any element f F obtaining elements A = f A and a × 0 c × a 0 ∈ ∈ C = f C instead of A and C . This new choice of elements may change the fields L c 0 0 0 1 and L but the newfieldswill still be Heisenbergextensions containing F(√p a,√p b) and 2 F(√p b,√p c) respectively. The next lemma will provide us with a suitable modification of A and C . From the proof of Theorem 3.7 we shall see that the compositum of 0 0 10 JÁNMINÁCˇ ANDNGUYỄNDUYTÂN these modified Heisenberg extensions can indeed be embeddedinto a Galoisextension M/F with Gal(M/F) U (F ). This explains our comment in the introduction in the 4 p ≃ paragraph related to the "automatic realization ofGaloisgroups". Lemma3.4. Assumethat thereexistC ,C E such that 1 2 × ∈ σ (C ) C a 1 2 B = . C σ (C ) 1 c 2 Then N (C )/A and N (C )/C are in F . Moreover, if we let A = N (C ) F(√p a) σc 1 0 σa 2 0 × σc 1 ∈ × and C = N (C ) F(√p c) , thenthereexists δ E suchthat σa 2 ∈ × ∈ × σ (δ) c p = AC− , δ 1 σ (δ) a p = CC− . δ 2 Proof. By Lemma3.2, we have σ (A ) σ (C ) C σ (N (C )) a 0 = N (B) = N a 1 N 2 = a σc 1 . A σc σc C σc σ (C ) N (C ) 0 (cid:18) 1 (cid:19) (cid:18) c 2 (cid:19) σc 1 Thisimpliesthat N (C ) N (C ) σc 1 = σ σc 1 . a A A 0 (cid:18) 0 (cid:19) Hence N (C ) σc 1 F(√p c) F(√p a) = F . × × × A ∈ ∩ 0 ByLemma3.2,we have σ (C ) C σ (C ) σ (N (C )) c 0 = N (B 1) = N 1 N c 2 = c σa 2 . C σa − σa σ (C ) σa C N (C ) 0 (cid:18) a 1 (cid:19) (cid:18) 2 (cid:19) σa 2 Thisimpliesthat N (C ) N (C ) σa 2 = σ σa 2 . c C C 0 (cid:18) 0 (cid:19) Hence N (C ) σa 2 F(√p a) F(√p c) = F . × × × C ∈ ∩ 0 Clearly,one has p Nσa(CC2− ) = 1, p Nσc(AC1− ) = 1.