Luận văn thạc sĩ galois cohomology and its applications in the embedding problem

MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Le Khac Nhuan GALOIS COHOMOLOGY AND ITS APPLICATIONS IN THE EMBEDDING PROBLEM MASTER THESIS IN MATHEMATICS Hanoi - 2021 BỘ GIÁO DỤC VIỆN HÀN LÂM VÀ ĐÀO TẠO KHOA HỌC VÀ CÔNG NGHỆ VN HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ Lê Khắc Nhuận ĐỐI ĐỒNG ĐIỀU GALOIS VÀ ỨNG DỤNG VÀO BÀI TOÁN NHÚNG Chuyên ngành: Đại số Lý thuyết số Mã số: 8460104 LUẬN VĂN THẠC SĨ NGÀNH TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: PGS TS Nguyễn Duy Tân Hà Nội - 2021 MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Le Khac Nhuan GALOIS COHOMOLOGY AND ITS APPLICATIONS IN THE EMBEDDING PROBLEM Major: Algebra and Number theory Code: 8460104 MASTER THESIS IN MATHEMATICS SUPERVISOR: Assoc Prof Nguyen Duy Tan Hanoi - 2021 Declaration I declare that this thesis titled ”Galois cohomology and its applications in the embedding problem” and the work presented in it are my own Wherever the work of others are involved, every effort is made to indicate this clearly, with due reference to the literature I confirm that this thesis has not been previously included in a thesis or dissertation submitted for a degree or any other qualification at this graduate university or any other institution I will take responsibility for the above declaration Hanoi, 20th September 2021 Signature of Student Le Khac Nhuan Acknowledgement First of all, I would like to express my deepest gratitude to Assoc Prof Nguyen Duy Tan was willing to supervise me despite my impoverished undergraduate mathematical background Thanks to his topic suggestion and his supervision, I have learnt so many things about Galois cohomology in particular and mathematics in general His guidance helped me all the time of writing this thesis I could not have imagined having a better supervisor for my MSc study Secondly, I would like to thank my lecturers, especially Dr Nguyen Dang Hop and Dr Nguyen Chu Gia Vuong, for many valuable mathematical lessons and insights I would like also to thank the Graduate University of Science and Technology and the Institute of Mathematics for providing a fertile mathematical environment Special thanks goes to Nguyen Van Quyet and Quan Thi Hoai Thu who all helped me in numerous ways during my study in Hanoi To Quyet in particular I owe you one for being a supportive friend as well as keeping me feel accompanied and conversations on various issues besides mathematics during my leisure times I would like also to thank my girlfriend Pham Thi Minh Tam, who has been by my side throughout my MSc, for all her love and support Last but not least, my thanks goes to my parents and siblings for always believing in me and encouraging me to follow my dreams Table of Contents Declaration Acknowledgement Table of Contents List of Symbols Introduction Profinite Groups 10 1.1 Infinite Galois Theory 10 1.2 Profinite Groups 16 Cohomology of Profinite Groups 30 2.1 Cohomology Groups 30 2.2 Functoriality 39 2.3 Universal Delta-functors 54 2.4 Induced Modules 62 2.5 Cup Products 68 2.6 Nonabelian Cohomology 74 Galois Cohomology 86 3.1 Algebraic Affine Group Schemes 86 3.2 Galois Cohomology 91 3.3 Galois Descent 99 The Embedding Problem 108 4.1 Group Extensions 109 4.2 The Embedding Problem 115 Conclusion 120 Reference 120 List of Symbols (G : H) index of a subgroup H in a group G 1A , id identity homomorphism A−B difference of a set B from a set A A∈A a object A in a category A A⊗B tensor product of abelian groups A and B A⋊G semidirect product of A by G AG subgroup of fixed points of A under the action of G Aα twisted group of A using cocycle α EXT (G, A) set of equivalence classes of group extensions of A by G Gk absolute Galois group of a field k H3 (Z/pZ) Heisenberg group of degree p Ks separable closure of a field K MGH (A) induced module of a H -module A T Fx (Ω/K) set of K -equivalence of Ω/K -twisted forms of x T ORS(A) set of isomorphisms of A-torsors ασ image of σ under a cocycle α Aut(A) group of automorphisms of a group A ∪ cup product δ, δ n connecting homomorphism Gal(Ω/K) Galois group of a Galois extension Ω/K Z[G0 ] group ring of G0 over Z L set of finite Galois subextensions of a Galois extension N set of normal open subgroups of a profinite group E(F, ϕ, π) embedding problem for a profinite group F with respect to epimorphisms ϕ and π µn (Ω) group of nth roots of unity in a field Ω ΩH fixed field of a subgroup H Ω× multiplicative group of a field Ω K algebraic closure of a field K σa, σa image of a under the action of σ StabG (a) stabilizer of a in G GLn (Ω) general linear group of degree n over a field Ω lim Ai − direct limit lim Gi − inverse limit c1/p pth root of c f∗ homomorphism of cohomology groups with respect to f k[G] representing k -algebra of an affine group scheme G over k B n (G, A) set of homogeneous n-coboundaries of a G-module A C n (G, A) set of homogeneous n-cochains of a G-module A H n (G, A) nth cohomology group of G with coefficient in A Z n (G, A) set of homogeneous n-cocycles of a G-module A n B (G, A) n C (G, A) set of inhomogeneous n-coboundaries of a G-module A set of inhomogeneous n-cochains of a G-module A n Z (G, A) ∂, ∂ n set of inhomogeneous n-cocycles of a G-module A inhomogeneous coboundary operator ∂, ∂ n coboundary operator inf inflation res restriction Ab category of abelian groups k-Alg category of unital commutative k -algebras Ek category of field extensions over k Grp category of groups Mod(G) category of G-modules Set∗ category of pointed sets Set category of sets Mod(G0 ) category of Z[G0 ]-modules coker(f ) cokernel of a homomorphism f im(f ) image of a homomorphism f ker(f ) kernel of a homomorphism f µn nth root of unity group scheme GLn general linear group scheme Ga additive group scheme Gm multiplicative group scheme SLn special linear group scheme 108 Chapter The Embedding Problem Since the early 19th century, the inverse Galois problem has been considered as one of the classical problems in Galois theory which is still unsolved This problem asks whether every finite group can be realized as a Galois group of some Galois extension over Q Naturally, the same question can be asked for other fields than Q More generally, let k be a field, and let L/k be a Galois extension with Galois group G Suppose that H is a profinite group and π : H − G is an epimorphism, i.e., π is a surjective continuous homomorphism One may ask the following question: is there any Galois extension Ω over L such that the Galois group Gal(Ω/k) is isomorphic to H and the following diagram is commutative Gal(Ω/k) // Gal(L/k)  π H // G, where the top arrow is the natural restriction, and the left arrow is the isomorphism between Gal(Ω/k) and H ? It should be observed that if H is a finite discrete group and L = k , i.e., G is trivial, then this question becomes the inverse Galois problem For every field K , we denote by GK its absolute Galois group It follows at once from Theorem 1.1.9.(2) that the existence of an epimorphism GK − S is equivalent to the existence of a Galois extension E over K such that Gal(E/K) is isomorphic to S Thus, one verifies easily that the above question can be reformulated as follows: is there any epimorphism ψ : Gk − H such that the following diagram is commutative Gk ψ H ϕ π G, 109 where ϕ is the natural restriction map? This reformulation leads to the embedding problem for a profinite group Our aim in this final chapter is to introduce the embedding problem for a profinite group, and give some applications of Galois cohomology in this problem We start with an interpretation of the second cohomology group in terms of group extensions, and then use this interpretation to prove Hoechsmann’s theorem in Section 4.2 The materials in this chapter have been taken from [7] and [6] 4.1 Group Extensions Before the development of group cohomology, the 2-cocycles have been known as factor sets and had a close connection with group extensions (c.f [19], pages 178-188) The aim of this section is to give a more concrete description of this connection in the case of profinite groups Throughout this section, we assume that G is a profinite group, and A is a finite G-module (we write A multiplicatively) Definition 4.1.1 A group extension of A by G is an exact sequence of profinite groups E: − ι A− ˆ −π G − G such that ι(σa) = σ ˆ · ι(a) · σ ˆ −1 , ˆ is a preimage of σ under π for σ ∈ G, a ∈ A and σˆ ∈ G It should be remarked that the above definition does not depend on the choice of the preimage σˆ of σ undet π Example 4.1.2 The Cartesian product A × G together with the operation given by (a, σ) · (b, τ ) = (a · (σb), στ ) for a, b ∈ A and σ, τ ∈ G is a group whose identity element is (1, 1), and where (a, σ)−1 = (σ −1 a−1 , σ −1 ) We denote this group by A ⋊ G One verifies at once that A ⋊ G endowed with the product topology is a profinite group that we call the semidirect product of A by G We thus obtain an exact sequence of profinite groups E1 : − ι A− π A⋊G− G− 1, 110 where ι(a) = (a, 1) and π(a, σ) = σ Then E1 is a group extension of A by G since for any σ ∈ G, a ∈ A and for any preimage (b, σ) of σ under π , we have (b, σ) · ι(a) · (b, σ)−1 =(b · (σa), σ) · (σ −1 b−1 , σ −1 ) =(b · (σa) · b−1 , 1) =ι(σa) Definition 4.1.3 Let E: − ι A− ˆ −π G − G E′ : − and ι′ A− ′ ˆ ′ −π G − G be group extensions of A by G We say that E and E ′ are equivalent if there exists ˆ − an isomorphism of topological groups f : G ˆ ′ such that the following diagram G is commutative // A ι // ˆ G π // G // f // A ι  ′ // ˆ′ G π′ // G // We denote by EXT (G, A) the set of equivalence classes of group extensions of A by G It is thus a pointed set, whose base point is the equivalence class of the group extension E1 defined in Example 4.1.2 The following theorem, due to Schreier (c.f [20]), gives an interpretation of H (G, A) in terms of group extensions Theorem 4.1.4 There is a bijection of pointed sets H (G, A) ∼ = EXT (G, A) In order to prove this theorem, we need some auxiliary results Let π : X − Y be a continuous surjection of topological spaces By a continuous section of π , we mean a continuous map s : Y − X such that π ◦ s = 1Y We state without proof the following lemma (c.f Proposition 2.2.2, page 29, [2]) Lemma 4.1.5 Let G be a profinite group, and let H be a closed normal subgroup of G Let π : G − s : G/H − G/H be the natural projection There exists a continuous section G of π such that s(1H) = The following corollary is straightforward from the above lemma 111 Corollary 4.1.6 Let π : G − G′ be an epimorphism of profinite groups Then there exists a continuous section s : G′ − G of π such that s(1) = By a normalized 2-cocycle of G-module A, we mean a 2-cocycle α ∈ Z (G, A) such that α(σ, 1) = α(1, σ) = Lemma 4.1.7 Every cohomology class in H (G, A) is represented by some normalized 2-cocycle Proof Let c be a cohomology class in H (G, A), and let α be a 2-cocycle representing c, so that σα(τ, ρ) · α(σ, τ ρ) = α(στ, ρ) · α(σ, τ ) It follows that σα(1, 1) = α(σ, 1) Since the map β : G − α(1, ρ) = α(1, 1) and A given by β(σ) = α(1, 1) is a continuous map, we obtain a 2-coboundary ∂ β given by (∂ β)(σ, τ ) = σβ(τ ) · β(στ )−1 · β(σ) Considering the 2-cocyle α′ = α · (∂ β)−1 , it is clear that α and α′ are cohomologous Moreover, we have α′ (σ, 1) = α(σ, 1) · (σα(1, 1))−1 = and α′ (1, σ) = α(1, σ) · α(1, 1)−1 = 1, hence α′ is a normalized 2-cocycle in c This proves our lemma We are now ready to prove Theorem 4.1.4 Proof of Theorem 4.1.4 We first define a map Φ : H (G, A) − EXT (G, A) as follows: for each cohomology class c ∈ H (G, A), let α be a normalized 2-cocycle representing c The existence of such normalized 2-cocycle follows from Lemma 4.1.7 Thus, we have σα(τ, ρ) · α(σ, τ ρ) = α(στ, ρ) · α(σ, τ ) and α(σ, 1) = α(1, σ) = The Cartesian product A × G together with the operation given by (a, σ) · (b, τ ) = (α(σ, τ ) · a · (σb), στ ) for a, b ∈ A and σ, τ ∈ G is a group whose identity element is (1, 1), and where (a, σ)−1 = (σ −1 α(σ, σ −1 )−1 · σ −1 a−1 , σ −1 ) 112 We denote this group by A ⋊α G One verifies at once that A ⋊α G endowed with the product topology is a profinite group, hence we obtain an exact sequence of profinite groups Eα : − ι π A −α A ⋊α G −α G − 1, where ι(a) = (a, 1) and π(a, σ) = σ An easy calculation shows that Eα is a group extension of A by G We then define Φ(c) to be the equivalence class of Eα in EXT (G, A) The image Φ(c) does not depend on the choice of the normalized 2-cocycle α Indeed, suppose that α′ is another normalized 2-cocycle representing c and its corresponding group extension is Eα′ : − ι π ′ ′ A −α A ⋊α′ G −α G − Then α and α′ are cohomologous, i.e., there exists β ∈ C (G, A) such that α = α′ ·∂ β , hence is is easily seen that β(1) = since α and α′ are normalized 2-cocycles One A ⋊α′ G given by f (a, σ) = (β(σ) · a, σ) verifies at once that the map f : A ⋊α G − is an isomorphism of topological groups making the following diagram commutative // A ια // A ⋊α G πα // G // G // 1, f // A ι α′  // A ⋊α′ G πα ′ // hence Eα and Eα′ are equivalent, i.e., the equivalence class of E coincides with the equivalence class of Eα′ , as desired It is easily seen that Φ is a map of pointed sets We next define a map Ψ : EXT (G, A) − H (G, A) as follows: for each equivalence class E ∈ EXT (G, A), let E: − ι A− ˆ −π G − G be a group extension of A by G representing E We choose a continuous section s of π such that s(1) = The existence of such continuous section follows from Corollary 4.1.6 From the definition of group extensions of A by G, it follows that for any σ ∈ G and a ∈ A we have ι(σa) = s(σ) · ι(a) · s(σ)−1 , i.e., s(σ) · ι(a) = ι(σa) · s(σ) (4.1) 113 For each pair σ, τ ∈ G, it is clear that π(s(σ) · s(τ )) = π(s(στ )) since s is a continuous section of π , hence there exists a unique α(σ, τ ) ∈ A such that s(σ) · s(τ ) = ι(α(σ, τ )) · s(στ ) The map α : G × G − A given by (σ, τ ) (4.2) α(σ, τ ) is a continuous, as one verifies at once Since ι is injective and s(1) = 1, it follows at once from (4.2) that α(σ, 1) = α(1, σ) = Moreover, for any σ, τ, ρ ∈ G, we have s(σ) · (s(τ ) · s(ρ)) =s(σ) · [ι(α(τ, ρ)) · s(τ ρ)] =[s(σ) · ι(α(τ, ρ))] · s(τ ρ) =[ι(σα(τ, ρ)) · s(σ)] · s(τ ρ) (from (4.1)) =ι(σα(τ, ρ)) · ι(α(σ, τ ρ)) · s(στ ρ), and (s(σ) · s(τ )) · s(ρ) =[ι(α(σ, τ )) · s(στ )] · s(ρ) =ι(α(σ, τ )) · ι(α(στ, ρ)) · s(στ ρ) Thus, it follows from the injectivity of ι that σα(τ, ρ) · α(σ, τ ρ) = α(σ, τ ) · α(στ, ρ), showing that α is a normalized 2-cocycle in Z (G, A) We then define Ψ(E) to be the cohomology class of α in H (G, A) The image Ψ(E) does not depend on the choice of the group extension E and the continuous section s Indeed, suppose that E′ : − ι′ ′ ˆ ′ −π G − G A− is another group extension representing E and s′ is a continuous section of π ′ satisfying s′ (1) = 1, so that for any σ ∈ G and a ∈ A, we have s′ (σ) · ι′ (a) = ι′ (σa) · s′ (σ) (4.3) Moreover, we have E and E ′ are equivalent, hence there exists an isomorphism of ˆ− topological groups f : G ˆ ′ making the following diagram commutative G A // ι // ˆ G π // G // f // A ι  ′ // ˆ′ G π′ // G // 114 It is easily seen that s′′ = f ◦ s is a continuous section of π ′ such that s′′ (1) = An easy calculation shows that α is the normalized 2-cocycle corresponding to the group extension E ′ and the continuous section s′′ Since π ′ (s′′ (σ)) = σ = π ′ (s′ (σ)), there exists a unique β(σ) ∈ A such that s′′ (σ) = ι′ (β(σ)) · s′ (σ) for σ ∈ G Then the map β: G − A given by σ β(σ) is a continuous map, i.e., β ∈ C (G, A) Let α′ be the normalized 2-cocycle corresponding to E ′ and s′ For any σ, τ ∈ G, we have s′′ (σ) · s′′ (τ ) =[ι′ (β(σ)) · s′ (σ)] · [ι′ (β(τ )) · s′ (τ )] =ι′ (β(σ)) · [s′ (σ) · ι′ (β(τ ))] · s′ (τ ) =ι′ (β(σ)) · [ι′ (σβ(τ )) · s′ (σ)] · s′ (τ ) (from (4.3)) =ι′ (β(σ)) · ι′ (σβ(τ )) · ι′ (α′ (σ, τ )) · s′ (στ ), and s′′ (σ) · s′′ (τ ) =ι′ (α(σ, τ )) · s′′ (στ ) =ι′ (α(σ, τ )) · ι′ (β(στ )) · s′ (στ ) Thus, it follows from the injectivity of ι′ that α(σ, τ ) · β(στ ) = β(σ) · σβ(τ ) · α′ (σ, τ ), hence α = ∂ β · α′ , i.e., α and α′ are cohomologous, showing that the cohomology class of α coincides with the cohomology class of α′ , as desired Finally, it remains to show that Ψ is the inverse of Φ Let c ∈ H (G, A) be a cohomology class, and let α be a normalized 2-cocycle representing c We then obtain a group extension Eα : − π ι A −α A ⋊α G −α G − defined as above One verifies at once that the map s : G − A ⋊α G given by s(σ) = (1, σ) is a continuous section of π such that s(1) = (1, 1) An easy calculation shows that α is the 2-cocycle corresponding to the group extension Eα and the continuous section s, showing that Ψ is the right inverse of Φ Conversely, let E ∈ EXT (G, A) be an equivalence class, and let E: − ι A− ˆ −π G − G be a group extension representing E We choose a continuous section s of π such that s(1) = Let α be the 2-cocycle corresponding to the group extension E and the continuous section s, so that we obtain a group extension of A by G Eα : − ι π A −α A ⋊α G −α G − 115 defined as above One verifies at once that the map f : A ⋊α G − ˆ given by G f (a, σ) = ι(a) · s(σ) is an isomorphism of topological groups making the following diagram commutative // A ια // A ⋊α G πα G // // f //  ι A // π ˆ G // G // 1, hence E and Eα are equivalent, showing that Ψ is the left inverse of Φ This concludes the proof Definition 4.1.8 A split group extension of A by G is a group extension E: − ι A− ˆ −π G − G for which there exists a continuous section s of π which is also a group homomorphism It is clear that the group extension E1 , defined in Example 4.1.2, is a split group extension of A by G since the map s : G − A ⋊ G given by s(σ) = (1, σ) is a continuous section of π which is also a group homomorphism Remark 4.1.9 From the proof in Theorem 4.1.4, one verifies at once that the equivalence class of E1 consists of all split group extensions of A by G, and this equivalence class corresponds to the trivial cohomology class in H (G, A) 4.2 The Embedding Problem In this section, we introduce the embedding problem for a profinite group and give a proof of Hoechsmann’s theorem The last part of this section is devoted to discussion of Heisenberg extensions Definition 4.2.1 (1) An embedding problem E(F, ϕ, π) for a profinite group F is a diagram F ϕ A H π G 1, with a short exact sequence of profinite groups and an epimorphism ϕ 116 (2) An embedding problem E(F, ϕ, π) is said to be solvable if there exists a continH such that π ◦ ψ = ϕ In this case, ψ is called uous homomorphism ψ : F − a solution of E(F, ϕ, π) A solution ψ is said to be proper if ψ is surjective We now investigate the embedding problem E(F, ϕ, π) F ϕ E: ι A H π G 1, for which A is a finite G-module and the short exact sequence is a group extension E of A by G Recall from Example 2.1.4.(5) that A is a F -module under the following action σa = ϕ(σ)a for σ ∈ F and a ∈ A It is clear that the homomorphims ϕ and 1A are compatible, hence we obtain an induced homomorphism of cohomology groups ϕ∗ : H (G, A) − H (F, A) The following theorem, due to Hoechsmann (c.f [21]), gives a characterization of solvability of the embedding problem E(F, ϕ, π) Theorem 4.2.2 Under assumptions as above, let ε ∈ H (G, A) be the cohomology class corresponding to the equivalence class of the group extension E (see Theorem 4.1.4) Then the embedding problem E(F, ϕ, π) is solvable if and only if ϕ∗ (ε) is trivial π Proof Let (H ×G F, ρ1 , ρ2 ) be the fiber product of the diagram H − G ϕ − F More precisely, H ×G F = {(h, σ) ∈ H × F | π(h) = ϕ(σ)} (4.4) and ρ1 , ρ2 are canonical projections onto H, F , respectively We thus obtain a commutative diagram A // ι′ ρ2 ×G F // H // F ρ1 // A ι //  H // ϕ π //  G // 117 with exact rows, where ι′ is given by ι′ (a) = (ι(a), 1) The upper exact sequence is a group extension of A by F since for any σ ∈ F, a ∈ A and a preimage (h, σ) of σ under ρ2 , we have (h, σ) · ι′ (a) · (h, σ)−1 =(h · ι(a) · h−1 , 1) =(ι(ϕ(σ)a), 1) (from (4.4)) =ι′ (σa) We denote this group extension by E ′ Choose a continuous section s of π such that s(1) = and let α be the the corresponding 2-cocycle of the group extension E and s, so that ε is represented by α, and ϕ∗ (ε) is represented by the cocycle ϕ∗ (α) = α ◦ ϕ2 given by ϕ∗ (α)(σ, τ ) = α(ϕ(σ), ϕ(τ )) It is easily seen that the map s′ : F − H ×G F given by s′ (σ) = (s(ϕ(σ)), σ) is a continuous section of ρ2 satisfying s′ (1) = (1, 1) since s(1) = One verifies at once that ϕ∗ (α) is the corresponding 2-cocycle of the group extension E ′ and s′ , hence the cohomology class ϕ∗ (ε) corresponds to the equivalence class of the group extension E ′ Now suppose that ϕ∗ (ε) is trivial It follows from Remark 4.1.9 that E ′ is a split group extension, i.e., there exists a continuous section t of ρ2 which is also a group homomorphism It follows at once that ψ = ρ1 ◦ t is a solution of the embed- ding problem E(F, ϕ, π) Conversely, suppose that ψ is a solution of the embedding problem E(F, ϕ, π) One verifies at once that the map t : F − H ×G G given by t(σ) = (ψ(σ), σ) is a continuous section of ρ2 which is also a group homomorphism, i.e., E ′ is a split group extension, hence Remark 4.1.9 shows that ϕ∗ (ε) is trivial, as desired This concludes the proof We end this section with a discussion of Heisenberg extensions We first introduce the Heisenberg group Let p be an odd prime The Heisenberg group of degree p, denoted by H3 (Z/pZ), is the group of × upper triangular matrices with entries in Z/pZ and 1’s in the diagonal, under the operation of matrix multiplication For every x, y, z ∈ Z/pZ, it is convenient to write (x, y, z) for the matrix  x z    0 y  ∈ H3 (Z/pZ)   0 Then the multiplication in H3 (Z/pZ) is given by (x, y, z) · (x′ , y ′ , z ′ ) = (x + x′ , y + y ′ , z + z ′ + xy ′ ) 118 We assume that k is a field of characteristic not p containing the group µp of pth roots of unity Let a and b be elements in k × which are p-independent, i.e., the classes of a, b are linearly independent in the Z/pZ-vector space k × /k ×p For every c ∈ k × , we let c1/p denote a pth root of c Then the field L = k(a1/p , b1/p ) is a Galois extension over k We choose a primitive pth root of unity ζ and define the generators σa and σb of Gal(L/k) by σa : a1/p ζa1/p , b1/p b1/p , σb : a1/p a1/p , b1/p ζb1/p The Galois group Gal(L/k) can be identified with Z/pZ × Z/pZ by letting σa corre- spond to (1, 0) and σb correspond to (0, 1) Consider the epimorphism π : H3 (Z/pZ) − Z/pZ × Z/pZ (x, y, z) − given by (x, y) We then obtain the embedding problem E(Gk , ϕ, π) for the absolute Galois group Gk Gk ϕ Z/pZ ι π H3 (Z/pZ) where ϕ is the natural epimorphism Gk − by ι(z) = (0, 0, z) Z/pZ × Z/pZ 0, Gal(L/k) = Z/pZ × Z/pZ, and ι is given We now identify µp with the group Z/pZ via ζ Recall from Corollary 3.3.9 that the Kummer exact sequence is the following: 0− Z/pZ − u ks× − ks× − 1, where u is given by u(x) = xp From Theorem 2.2.16, we obtain the connecting homomorphism δ : k× − H (Gk , Z/pZ) It should be remarked that Gk acts trivially on Z/pZ since k contains µp , hence the group H (Gk , Z/pZ) consists of continuous homomorphisms from Gk to Z/pZ For any c ∈ k × , we let χc denote its image under δ From Remark 2.2.19.(1), let us clarify the definition of the character χc as follows: let σ be an element in Gk , so that σ(c1/p )/c1/p is a pth root of unity; we may assume that σ(c1/p )/c1/p = ζ r Then χc maps σ to r + pZ ∈ Z/pZ Thus, one verifies at once that the epimorphism ϕ can be represented by ϕ(σ) = (χa (σ), χb (σ)) for σ ∈ Gk Recall from Example 2.5.2.(1) that the multiplication in the ring Z/pZ is a pairing from Z/pZ × Z/pZ to Z/pZ, hence we obtain a cup product ∪ : H (Gk , Z/pZ) × H (Gk , Z/pZ) − H (Gk , Z/pZ) 119 The following proposition shows that the cup product χa ∪ χb is the obstruction to solvability of the embedding problem E(Gk , ϕ, π) Proposition 4.2.3 Under the above assumptions, the embedding problem E(Gk , ϕ, π) is solvable if and only if the cohomology class of χa ∪ χb is trivial Proof Consider Z/pZ as a Z/pZ × Z/pZ-module under the trivial action It is easily seen that ι(Z/pZ) lies in the center of H3 (Z/pZ), hence the short exact sequence 0− ι Z/pZ − π H3 (Z/pZ) − Z/pZ × Z/pZ − is a group extension of Z/pZ by Z/pZ × Z/pZ that we denote by E Then the map s : Z/pZ × Z/pZ − H3 (Z/pZ) given by s(x, y) = (x, y, 0) is a continuous section of π Let α be the 2-cocycle corresponding to the group extension E and s From Theorem 4.2.2, it suffices to show that the cocycle ϕ∗ (α) = α ◦ ϕ2 coincides with the cocycle χa ∪ χb Indeed, from the proof of Theorem 4.1.4, for any u = (x, y), v = (x′ , y ′ ) ∈ Z/pZ × Z/pZ, we have s(u) · s(v) = ι(α(u, v)) · s(u + v), i.e., (x, y, 0) · (x′ , y ′ , 0) = (0, 0, α(u, v)) · (x + x′ , y + y ′ , 0), hence α(u, v) = xy ′ This shows that ϕ∗ (α)(σ, τ ) = α(ϕ(σ), ϕ(τ )) = χa (σ)χb (τ ) since ϕ(σ) = (χa (σ), χb (σ)) and ϕ(τ ) = (χa (τ ), χb (τ )) On the other hand, from the defini- tion of cup products (see Section 2.5), we have (χa ∪ χb )(σ, τ ) = χa (σ)(σχb (τ )) = χa (σ)χb (τ ) since Gk acts trivially on Z/pZ, hence ϕ∗ (α) = χa ∪ χb , as desired This concludes the proof For a generalization of the above proposition, we refer the reader to Sharifi’s dissertation [22] 120 Conclusion In this thesis, we have presented the following: • The concept of profinite groups and its basic properties We also give a proof of the fundamental theorem of infinite Galois theory and show that every Galois group is a profinite group • The general constructions of cohomology of profinite groups We also introduce some important concepts, including universal δ -functors, induced modules, cup products, torsors, twisting, and use them to prove some elementary results • The functorial approach of Galois cohomology We also introduce Galois descent and give some typical examples of algebraic affine group schemes • The basic concepts of the embedding problem for a profinite group We also give a proof of Hoechsmann’s theorem and discuss Heisenberg extensions of degree p3 121 Bibliography [1] Patrick Morandi, 1996, Field and Galois Theory, Springer-Verlag, New York [2] Luis Ribes, Pavel Zalesskii, 2000, Profinite Groups, Springer, Berlin, Heidelberg [3] Stephen Willard, 1970, General Topology, 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ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Le Khac Nhuan GALOIS COHOMOLOGY AND ITS APPLICATIONS IN THE EMBEDDING PROBLEM Major: Algebra and Number theory... MASTER THESIS IN MATHEMATICS SUPERVISOR: Assoc Prof Nguyen Duy Tan Hanoi - 2021 Declaration I declare that this thesis titled ? ?Galois cohomology and its applications in the embedding problem? ?? and the. .. with infinite Galois theory and aim to prove the fundamental theorem of infinite Galois theory Then we give some characterizations of profinite groups and show that every Galois group is a profinite