Skew Group Rings and Galois Theory for Skew Fields 1

Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 2, 49-62 Skew Group Rings and Galois Theory for Skew Fields 1 Xu Yonghua Institute of Mathematics Fudan University, Shanghai, China K. P. Shum Department of Mathematics The University of Hong Kong Pokfulam Road,Hong Kong, China (SAR) kpshum@maths.hku.hk Abstract In this paper,we study the Galois theory for the skew field F with a finite outer group G of automorphisms of F. For such an outer group G, we prove the Galois fundamental theorem which can be stated as the same as the classical Galois fundamental theorem for fields only by replacing the words subfield and skew subfield. Moreover, we provide a new method to prove the Fundamental theorem for Galois corresponding subgroups to subrings of L(P,F). Although the Galois fundamental theorem for skew field was proved previously by Jacobson and Cohn in the literature, our approach is somewhat different, in particular, we do not use any field theory but only use the techniques of complete rings of linear transformations developed by Xu in 1980. Mathematical subect classification: 16K40, 16A28, 16W40 Keywords: Skew fields; Galois Fundamental theorem; Semi-linear transformations 1 Introduction In this paper, we follow the notions and terminologies given in Jacobson [5] and Xu [8]. Our aim is to prove the fundamental theorem of Galois theory 1 This research is partially supported by a UGC(HK) grant #2160297 2006/07 and also partially supported by NSF grant (China) #19971028.

50 Xu Yonghua and K. P. Shum for a skew field F under the assumption that the group G of automorphisms of F is finite outer. In this case, the Galois fundamental theorem for skew field can be stated as the same as the classical Galois fundamental theorem for fields only by replacing the words subfield and skew subfield. The results of skew fields in the literature have been proved and collected by Jacobson and Cohn in their books 1] and [4] previously (see also [2] and [3]). However, our approach of establishing the Galois theory for skew fields is somewhat somewhat different from Jacobson and Cohn as we only use the techniques in complete rings of linear transformations developed by Xu in [9]. In particular, we shall demonstrate that the Galois fundamental theorem for skew fields can be re-established by using a different method without field theory. As a consequence, our results give a suggestion that the complicated Galois theory for differential algebras may perhaps be re-established without using Picard- Vessiot field extensions ( see [6] and [7]). For notations and terminologies not given in this paper, the reader is refereed to the Encyclopedia of Mathematics and its Applications by Cohn [1] and also the text of Jacobson [4]. 2 Preliminaries Let M = i Γ Fu i be a left vector space over a skew field F, where {u i } i Γ is a basis indexed by an arbitrary set Γ. We use L(F, M) = End F M to denote the complete ring of all F -linear transformations of F M. Then F M L(F,M) becomes a bimodule. If E(M) is the set of all endomorphisms of the additive group (M,+),then E(M) is certainly a ring, for which there arise two endomorphism rings, namely, the ring of left endomorphisms, denoted by E l (M) and the ring of right endomorphisms, denoted by E r (M). Obviously, for elements a l E l (M) and b r E r (M), a l b r = b r a l and E l (M) = (E r (M)) op. As usual, the bimodule F M L(F,M) is defined by the maps F F L E l (M) and L(F, M) L(F, M) r E r (M) so that mf l ω r = fmω for f l F L, ω L(F, M) and m M. It is clear that the subrings F L and L(F, M) r of E(M) are precisely the centralizers of each other. In this case, the rings F and L(F, M) are said to be the centralizers of each other in M. Let G be a group of automorphisms of F and ψ G. Then we can easily determine an F -semi-linear automorphism S of F M = i Γ Fu i by ψ. In fact, for a given invertible element ω in L(F, M), we can define a map S : i Γ f iu i i f ψ i (u iω) such that ( i f iu i )S = i f ψ i (u is) = i f ψ i (u iω). It is clear that S is an F -semi-linear automorphism of F M associated with ψ G. In order to indicate ψ more explicitly, we always write S =(S, ψ). It is also clear that there are possibly infinite many F -semi-linear automorphisms

Skew group rings and Galois theory 51 of F M associated with the same ψ G. We now let Θ be the set of all F -semi-linear automorphisms associated with the group G. Then,Θ is clearly a group and the multiplication of elements S and S in Θ is the same as the multiplication in E(M) since Θ E(M). Actually, one can easily see that Θ r E r (M). On the other hand, we can introduce a relation on Θ. We now define S S on Θ if S and S have the same associated isomorphism ψ G, that is, S =(S, ψ) S =(S,ψ). Obviously, the relation is an equivalence relation on Θ. Hence we can participate Θ into equivalence classes Θ ={ S S Θ}, where S = {S Θ S S} is determined by the associated isomorphism ψ of (S, ψ), that is, S =( S,ψ). It is easy to see that Θ is a group whose product is induced by Θ, that is, S1 S2 = S 1 S 2. Clearly, Θ = G is a group. As motivated by the above discussion, we formulate the following definition. Definition 2.1 Let G be a group of automorphisms of a skew field F. Then the set Θ={(S, ψ) ψ G} described above is called the group of F -semi-linear automorphisms of F M associated with G. We denote the group of equivalence classes of Θ associated with G by Θ. We next consider the L(F, M)-module ΘL(F, M) = S Θ SL(F, M) which is generated by Θ in E(M). We call ΘL(F, M) = S Θ SL(F, M) the smash product of the ring L(F, M) and the group Θ. We now formulate the following lemma. Lemma 2.2 The L(F, M)-module ΘL(F, M) = S Θ SL(F, M) is a ring with identity. Proof: (i) It is clear that SL(F, M) =L(F, M)S for S Θ. For the mapping I S defined by ω I S = SωS 1,ω L(F, M) is clearly a ring automorphism of L(F, M). (ii) We now claim that SL(F, M) =S L(F, M) if and only if S S for S, S Θ. In fact, if {u i } i Γ is a basis of F M = i Γ Fu i, then we can easily see that {u i S} i Γ and {u i S } i Γ are both F -basis of F M. Then there exists an invertible element l L(F, M) such that (u i S)l = u i S for all i Γ. Assume that S S. Then for f F, we have (fu i )Sl = f ψ (u i Sl) =f ψ (u i S )=(fu i )S, for all i Γ. This proves that Sl = S. Hence SL(F, M) =S L(F, M). The converse is clear and our claim holds. (iii) We now prove that S 1 L(F, M)S 2 L(F, M) =S 1 S 2 L(F, M), for S i Θ, i= 1, 2. In fact, for S i ω i S i L(F, M), i=1, 2, we have S 1 ω 1 S 2 ω 2 = S 1 S 2 ω I S 2 1 1 ω 2 S 1 S 2 L(F, M). This completes the proof.

52 Xu Yonghua and K. P. Shum Definition 2.3 Let G = {ψ } I be a group indexed by the set I and define Θ ={ S } I accordingly, where S =( S,ψ ) is the equivalence class of S = (S,ψ ) Θ, as described above. Then a subset B = {S } I of Θ is called a representative set of Θ associated with the group G if S i S for any i in I and (1 M, 1 F ) B, where 1 F =1 G G. By Lemma 2.2, we immediately obtain the following corollary. Corollary 2.4 The smash product ΘL(F, M) of L(F, M) and the group Θ can be written by ΘL(F, M) = BL(F, M) = I S L(F, M), where B = {S } I is a representative set of Θ associated with the group G = {ψ } I. Specially, if G = n, then the smash product ΘL(F, M) = BL(F, M) = G S L(F, M). Moreover, if B = {S } I is an another representative set of Θ, then S L(F, M) =S L(F, M) for I, and hence, BL(F, M) = B L(F, M). 3 Crucial Lemmas In this Section, we prove several crucial lemmas which are related to skew group rings and Galois theory. We first let F be a skew field and E(F ) the ring of all automorphisms of the additive group (F, +). We call an automorphism ψ of F inner if there exists a unit u of F such that ψ = I u = u l u 1 r, that is, f Iu = fu l u 1 r = ufu 1 for f F, and outer otherwise. Let G be a finite group of automorphisms over a skew field F. Then G is said to be inner if every ψ G is inner. On the other hand, G is said to be outer if 1 G is the only inner automorphism in G. Let G = {ψ 1,, ψ G } be a finite group of automorphisms of F. Then, we write InvG = {x F x ψ = x, ψ G}. Obviously, InvG is a skew subfield of G-invariants of F. We always denote it by P = InvG. As usual, we use the notation F/P to mean that P is a skew subfield of F. The notation P F means that F over P is a left space and [F : P ] L denotes the dimension of P F over P. Let (a i ) n m be a matrix over a skew field F. Then, a matrix (a i ) n m over a skew field F is said to be full rank if and only if (a i ) n m can be transformed into (a i ) n m by usual elementary operations, where a ii =1, a i =0, i > ; =1,,m; i =1,,n; n m. The following lemma given by Xu in [8] plays an important role in the study of Galois theory.

Skew group rings and Galois theory 53 Lemma 3.1 Let F M = i Γ Fu i be a vector space over a skew field F. Let y 1,,y n be n linearly independent elements of F M over F. Suppose that a 11 x 1 + + a 1m x m = y 1 a 21 x 1 + + a 2m x m = y 2 (3.1) a n1 x 1 + + a nm x m = y n is a system of linear equations with coefficients a i F and the system (3.1) has a solution in M. Then m n and the matrix (a i ) n m is full rank, and if we assume that (a i ) n n is full rank,for i, =1,,n, then for any given elements Y 1,,Y n of M (not necessarily F -independents) the system of linear equations n i=1 a ix = Y,,,n, has a solution in M and the solution can be written as X i = n b iy, i =1,,n, for b i F. Lemma 3.2 Let F be a skew field and G = {ψ 1,,ψ G } a finite group of automorphisms of F.LetP = Inv G, [F, P] L = the dimension of P F over P. Then (i) [F : P ] L G (ii) If F = [F :P ] L α=1 Pf α with basis {f α } α=1,,[f :P ]L over P, then the matrix (f ψ α ) [F :P ]L G is full rank, and hence the matrix (f ψ α ) [F :P ]L [F :P ] L is also full rank. Proof: We let F M = G +1 i=1 Fu i be a left vector space over F with a basis {u i } i=1,, G +1. Then it is clear that M = G +1 i=1 Fu i = i=1,, G +1 Pf α u i = α=1,,[f :P ] L i,α Pv(α) i, where v (α) i = f α u i, a vector space over P with a basis {v (α) i } i=1,, G +1. α=1,,[f :P ] L Denote the complete ring of linear transformations of left vector space P M = α,i Pv(α) i over P by L(P, P M). Following Section 1, we still denote B = {(S,ψ ) ψ G} as a representative set of Θ, and by Corollary 2.4, ΘL(F, M) = BL(F, M) = G S L(F, M) is a dense subring of L(P, M), since P =InvG. On the other hand, since P M = i,α Pv(α) i is a finite dimensional vector space over P, L(P, M) = G S L(F, M). By using the above notations, we can now prove (i). Suppose that [F : P ] L > G. Then we can pick G +1 elements v (1) i,,v ( G +1) i from the P -basis {v (1) i,,v ( G +1) i,,v ([F :P ] L) i } i=1,, G +1 of P M. It is clear that for any F -linearly independent elements, say, u 1,,u G +1,of F M = G +1 i=1 Fu i, there exists an element σ L(P, M) = G S L(F, M) such that v (α) i σ = u α for α =1,, G + 1. Write σ = G S ω,ω L(F, M),.

54 Xu Yonghua and K. P. Shum Then we have v (α) i σ = G v (α) i S ω = G f ψ α (u i S ω )= G a α x = u α, α =1,, G + 1 (3.2) where a α = f ψ α F, x = u i S ω M, for α =1,, G +1; =1,, G. Since the system (3.2) of linear equations with coefficients a α F has a solution x = u i S ω for =1,, G, and {u α } α=1,, G +1 is a basis of F M over F. Now by Lemma 3.1, G G + 1 holds and hence we arrive at a contradiction. This proves (i). (ii). From (i,) we have [F : P ] L G. Hence P M = i=1,, G +1 Pv (α) i α=1,,[f :P ] L with a basis {v (α) i }. Hence the system of equations (3.2) becomes G a α x = u α for α =1,, [F : P ] L. By Lemma 3.1 again, the matrix (a α ) α=1,,[f :P ] L,, G is full rank and thereby, (a α ) [F :P ]L [F :P ] L is also full rank, where a α = f ψ α. From now on, we consider the special case F M = F F. We first consider G = {ψ 1,,ψ G } which is a finite group of automorphisms of F and P =InvG. Now, by Lemma 3.2, [F : P ] L G. Now,we let [F : P ] L = n and P F = n α=1 Pf α, a left space over the skew subfield P of F with a basis {f α } α=1,,n. Let L(P, F) be the complete ring of P -linear transformations of P F over P. Then P and L(P, F) are precisely the centralizers of each other in F, that is, P L E l (F ) and L(P, F) E r (F ) are centralizers of each other in E(F ). On the other hand, we write the action of each ψ G on the space P F from the right side, that is, (fu)ψ = f ψ (uψ) for any elements f and u of F. Since P =InvG, (pu)ψ = p ψ u ψ = p(uψ) for all p P and u F. Hence ψ L(P, F) and G L(P, F),and consequently, G ψ F = GF L(P, F).This completes the proof. In the sequel, we always assume that the finite group G is outer. Lemma 3.3 Let G = {ψ 1,,ψ G } be a finite outer group of automorphisms of skew field F and let P = Inv G. Then the following statements hold: (i) If ψ is an automorphism of F/P and ψ = l ψ g for ψ G, g F, =1,,l, then ψ {ψ 1,,ψ l }. (ii) L(P, F) =GF = G ψ F is a skew group ring, and dim F L(P, F) = dim L(P, F) F = G. (iii) G = Gal F/P.

Skew group rings and Galois theory 55 Proof: (i) By assumption, we write ψ = l ψ g, g F, =1,,l; l G. Since (fu)ψ = f ψ (uψ) for all f F and u F, it follows by ψ = l ψ g that l (f ψ f ψ )(uψ g ) = 0 (3.3) We now show that there exists an element ψ i {ψ 1,,ψ l } such that ψ = ψ i. Suppose on the contrary that ψ ψ i for i =1,,l. Then by using induction,we can start with ψ ψ l and assume that there exists an element f 0 F such that fo ψ f ψ l o.then we denote h(l) 1,,l 1. Now, from (3.3), for any u F, we have Hence, uψ l g l = h (l) (uψ g )= =(fo ψ f ψ l o ) 1 (f ψ o u(ψ g h (l) L ) f ψ o ),= ψ l g l = ψ g h (l) L (we may assume that h (l) l 1 0) (3.4) Acting ψ l g l on F F, then we have (fu)ψ l g l = f ψ l (uψl g l ), for all f F, u F. Thus, by (3.4) that is, Hence l 1 (fu)(ψ g h (l) L )= h (l) f ψ (uψ g )= (h (l) f ψ l (uψ g h (l) L, ) f ψ l h (l) (uψ ig i ) f ψ f ψ l h (l) )(uψ g ) = 0 (3.5) Now, suppose that h (l) l 1 f ψ l 1 f ψ lh (l) l 1 =0, for all f F. Then ψ l = ψ l 1 I (l) h, l 1 where I (l) h = h (l) l 1,L h(l) 1 l 1,r is inner. It is now easy to see that I = ψ 1 l 1 h (l) l 1 ψ l G. l 1 Since G is outer, I (l) h =1 G, and hence ψ l = ψ l 1. This clearly contradicts l 1

56 Xu Yonghua and K. P. Shum to ψ l f ψ l 1 h (l) l 1 h (l 1) ψ l 1. Thus, there exists an element f 1 F such that h (l) l 1 f ψ l 1 1 0. Repeating the above procedure by starting from (3.3) and let =(h (l) l 1 f ψ l 1 1 f ψ l 1 h(l) follows from (3.5) that l 1 ) 1 (h (l) f ψ 1 f ψ l 1 h(l) ), =1,,l 2. Then it l 2 ψ l 1 g l 1 = ψ g h (l 1) L (we may assume that h (l 1) l 2 0). (3.6) By comparing (3.4) with (3.6), we can see, by our induction hypothesis, that ψ 2 g 2 = ψ 1 g 1 h (2) 1L, 0 h(2) 1 F (3.7) Acting ψ 2 g 2 on F F, we have (fu)ψ 2 g 2 = f ψ 2 (uψ 2 g 2 ) for all f F, u F. Thus, that is, and (fu)ψ 1 g 1 h (2) 1L = f ψ 2 (uψ 1 g 1 h (2) 1L, ) h (2) 1 f ψ 1 (uψ 1 g 1 )=f ψ 2 h (2) 1 (uψ 1g 1 ) (h (2) 1 f ψ 1 f ψ 2 h (2) 1 )(uψ 1g 1 )=0. (3.8) We now consider the following cases: Case 1: if h (2) 1 f ψ 1 f ψ 2 h (2) 1 =0, for all f F, then ψ 2 = ψ 1 I (2) h,and hence, 1 ψ 2 = ψ 1 since G is outer. Since ψ 2 ψ 1 on G, we obtain a contradiction. Case 2: if there exists an element f 1 F such that h (2) 1 f ψ 1 1 f ψ 2 1 h(2) 1 0, then from (3.8), it follows that uψ 1 g 1 =0, for all u F, and hence ψ 1 g 1 =0. But ψ 1 g 1 0 by our assumption. This completes the proof of (i). (ii). We first note that GF = G ψ F is a free module over F. For, if G ψ g = 0, and g i 0, then ψ i = i ψ g gi 1. By using the above result (i), we have ψ i = ψ for some i. Clearly, this is impossible. Hence g i =0, for all i =1,, G. Next we note again that for f F and ψ G, ψ f = f ψ 1 ψ. In fact, (gu)ψ f = g ψ (uψ f)=(guf ψ 1 elements g, u in F. Hence ψ f = f ψ 1 )ψ =(gu)(f ψ 1 ψ ) for ψ. It is now clear that GF = G ψ F is a ring which is called a skew group ring, denoted GF by G F. Clearly, G F is a dense subring of L(P, F). By dim P F<, we obtain G F = L(P, F), as required. (iii). If ψ Gal F/P, then ψ L(P, F) and so ψ G by (i). This proves that Gal F/P = G. Thus, our proof is completed.

Skew group rings and Galois theory 57 Lemma 3.4 Let G be a finite outer group of automorphisms of F and P = Inv G. Then (i) C F (P )={f F fp = pf for p P } is the center of F. (ii) [F : P ] L = G = Gal F/P. Proof: (i) If f C F (P ), then the inner I f is an element of L(P, F), since (pu)i f = p(ui f ) for p P and u F. Hence I f = l ψ g and I f = ψ for some {1,,l} by Lemma 3.3 Since G is outer, I f =1 G. Hence gi f = g, that is, fgf 1 = g for all g F. This completes the proof. (ii) Let [F, P] L = n, F = n α=1 Pf α, where P =InvG. For any 0 u F, P F = P Fu = n α=1 Pf αu = n α=1 Pv(α), v (α) = f α u. Let σ L(P, F); v (α) σ = Y α, α =1,,n. By Lemma 3.3, σ = m ψ g, m G. Hence v (α) σ = m (f α u)ψ g = m f ψ α (uψ g )=Y α, α =1,,n (3.9) where g F, m G. Since (3.9) is a system of linear equations with coefficient f ψ α F, by Lemma 3.2, n G and therefore the matrix (f ψ α ) n G is full rank and consequently,the matrix (f ψ α ) n n is also full rank. We now consider the following n a α X = Y α, α =1,,n, a α = f ψ α, α, =1,,n (3.10) system of linear equations with coefficients a α = f ψ α F and Y α F. Since the matrix (a α ) n n is full rank, the system (3.10) has a solution X 1,,X n in F. On the other hand, for every X and uψ, we have h F such that (uψ )h = X, =1,,n. Putting σ = n ψ h, we have σ L(P, F), since ( n α=1 p αv (α) )ψ h = n α=1 p α(v (α) ψ h ), and hence p ψ = p for p P = Inv G. Acting σ on the P -basis {v (α) } α=1,,n, we have v (α) σ = n (f α u)ψ h = n f ψ α (uψ h )= n a α X = Y α = v (α) σ for α =1,,n. This proves that σ = σ. Hence L(P, F) = n ψ F, n G. Now, by Lemma 3.3 again, we have L(P, F) = G ψ F = n ψ F, and thereby G = n =[F : P ] L. This completes the proof.

58 Xu Yonghua and K. P. Shum Lemma 3.5 Let G = {ψ 1,,ψ G } be a finite outer group of automorphisms of F.LetP = Inv G and K be an intermediate skew subfield between P and F. Write L(K, F) as the complete ring of K-linear transformations of the skew field K F over K and let σ = I ψ g be an element of L(P, F). Suppose that σ is also an element of L(K, F). Then (i) all direct summands ψ g of σ are elements of L(K, F), for I (ii) If H = L(K, F) G, then H is a subgroup of G and L(K, F) =HF is a skew group subring of GF. Proof: For the sake of simplicity, we first write σ = l ψ g. Then, by assumption σ L(K, F), and hence (ku)σ = k(uσ) for all k K and u F, that is, l kψ (uψ g )= l k(uψ g ). Now, we have l (k ψ k)(uψ g ) = 0 for all k K, u F (3.11) We observe that if k ψ = k for all k K then ψ L(K, F). We now need to show that k ψ k =0, for all k K and all =1,,l. For this purpose, suppose that there exists an element, for example, ψ l of {ψ 1,,ψ l } such that ψ l / L(K, F). Then we proceed to show that there exists a contradiction. Since ψ l / L(K, F), there exists an element k o K such that k ψ l o k o 0. Writing a (l) =(k o k ψ l o ) 1 (k ψ o k o ), for =1,,l 1. Then formula (3.11) can be rewritten as Hence uψ l g l = a (l) (uψ g )= u(ψ g a (l) L ), for all u F. ψ l g l = ψ g a (l) L, a(l) F (3.12) It is clear that a (l),,..., l 1, are not all equal to 0. We may assume that a (l) l 1 0. Since ψ lg l L(P, F), (pu)ψ l g l = p(uψ l g l ) for all p P =InvG. It follows from (3.12) that l 1 a(l) p(uψ g )= l 1 p(a(l) uψ g ) for all p P, that is, (a (l) p pa(l) )(uψ g ) = 0 for p P (3.13)

Skew group rings and Galois theory 59 If a (l) p = pa(l) holds for all p P and = 1,..., l 1, then a (l) C F (P ) = the center of F by Lemma 3.4. Hence, a (l) = a (l) l 1 ψ g a (l) L F, and ψ lg l =. Applying Lemma 3.3, we have ψ l {ψ 1,..., ψ l 1 },however,this is impossible. Thus, there must be an element p o P such that a (l) l 1 p o p o a (l) l 1 for = l 1. Similar to the above procedure, we can write a (l 1) =(p o a (l) l 1 a(l) l 1 p o) 1 (a (l) p o p o a (l) ), for =1,..., l 2 Then, it follows from (3.13) that uψ l 1 g l 1 = l 2 uψ g a (l 1) L for all u F, that is, l 2 ψ l 1 g l 1 = ψ g a (l 1) L,a(l 1) F, =1,..., l 2. (3.14) Again, we may assume that a (l 2) l 2 0. Repeating the above procedure, we can show that a (l 1) are central elements of F for =1,..., l 2. Thus, from (3.14), we deduce that ψ l 1 {ψ 1,..., ψ l 2 }, a contradiction. Comparing (3.14) with (3.12) we can obtain the following equality by induction ψ 2 g 2 = ψ 1 g 1 a (1) 1L,a(1) 1 F (3.15) Since ψ 2 g 2 L(P, F), (pu)ψ 2 g 2 = p(uψ 2 g 2 ) for all p P. It follows that (a (1) 1 p pa (1) 1 )(uψ 1 g 1 ) = 0 for all p P and u F. (3.16) If a (1) 1 p = pa(1) 1 for all p P, then a (1) 1 C F (P ) is a central element of F. It follows from (3.15) and Lemma 3.3 that ψ 2 = ψ 1, but this is impossible. Thus, we can find an element p 1 P such that a (1) 1 p 1 p 1 a (1) 1 0 and hence, by equation (3.16),uψ 1 g 1 =0, for all u F. This shows that ψ 1 g 1 = 0,but since ψ 1 g 1 0, a contradiction. This completes the proof of (i). (ii). If σ L(K, F) then it is clear that σ L(P, F) =GF = G ψ F, by Lemma 3.3. Hence, σ = I ψ g, {ψ } I G. Now,by statement (i), {ψ } I L(K, F). This shows that {ψ } I H = G L(K, F) and σ HF and consequently, L(K, F) = HF. It remains to show that H is a group. However, the proof of this part is easy, since 1 G =1 F G L(K, F) and ψ i ψ H if ψ i,ψ are in H.( see Lemma 3.3.10 in [1]).

60 Xu Yonghua and K. P. Shum 4 Galois Fundamental theorem for skew fields We now formulate the Galois fundamental theorem for skew fields. Recall that F/P means that P is a skew subfield of F. By a skew subfield K of F/P, we mean that K is an intermediate skew field between F and P. An automorphism ψ of F/P means that ψ is an automorphism of F such that a ψ = a, for every a P. The set G of all automorphisms of F/P is called the Galois group of F over P, denoted by GalF/P. We now give the following definition. Definition 4.1 Let G be a finite outer group of of automorphisms of a skew field F and P = Inv G. Then a skew subfield K of F/P is called normal over P if K ψ = K, for every ψ Gal F/P. Theorem 4.2 (Galois Fundamental theorem for skew fields). Let F be a skew field and G a finite outer group of automorphisms of F. Let P = Inv G and Λ={H}, the set of subgroups of G. Denote the set of intermediate skew fields between F and P (the skew subfields of F/P) by Σ. Then the mappings H Inv H, K Gal F/K, H Λ, K Σ, are inverse mappings and they are also biective mappings of Λ onto Σ and of Σ onto Λ. Moreover, we have the following properties of pairing a) H 1 H 2 Inv H 1 Inv H 2 b) H =[F : Inv H] L, [G : H] =[Inv H : P ] L c) H is normal in G Inv H is normal over P. In this case, Gal (Inv H)/P = G/H. Proof: Let H Λ. Then H is a finite outer subgroups of G. Let K = Inv H. Then by applying Lemma 3.3 to H instead of G and applying Lemma 3.3 again to K =InvH instead of P =InvG. we can easily obtain that H = Gal F/K which is the Galois group of F over K. It is now clear that H Inv H, Inv H Gal F/(Inv H) =H, for all H Λ. On the other hand, let K Σ and write H = Gal F/K. We now show that Inv H = K. By Lemma 3.5, we have L(K, F) =(G L(K, F))F. Also, by Lemma 3.3, we have H = G L(K, F), since H = Gal F/K. Because K and L(K, F) are centralizers of each other in F, it is clear that Inv H = K. Hence, the mappings H Inv H, K Gal F/K are inverse mappings. It remains to prove (b), since (a) is clear from the above result. By Lemma 3.4, [F : P ] L = G and [F :InvH] L = H. Hence,it follows that [G : H] = [Inv H : P ] L, since H [G : H] =[F :InvH] L [Inv H : P ] L. c) Let H Λ and K =InvH. Assume that K is normal over P. Then by the above definition, K ψ = K for all ψ Gal F/P. By Lemma 3.3,

Skew group rings and Galois theory 61 G = Gal F/P. It is easy to see that ψ 1 Hψ = H for all ψ G. Hence H is a normal subgroup in G. Conversely, if H is a normal subgroup in G, then K ψ = K for all ψ G. Hence, for all ψ Gal F/P, K is normal over P, by Lemma 3.3. Finally we proceed to show that if K is normal over P, then Gal K/P = G/H. In fact, every ψ G maps K into itself and so the restriction ψ = ψ/k is an automorphism of K/P. Thus we have the restricted homomorphism ϕ : ψ ψ which maps G into Gal K/P. Since G = Gal F/P, by Lemma 3.3 again, we have Ḡ = Gal K/P. It is clear that the kernel of ϕ is the set ψ of G such that ψ/k =1 K. Hence, the kernel of ϕ is H. This proves that Ḡ = G/H = Gal (Inv H)/P. Thus the proof is completed. Theorem 4.3 ( A pairing theorem of subgroups to subrings of L(P, F) ). Let F be a skew field and G, a finite outer group of automorphisms of F. Write P = Inv G as before. Let Λ={H} be the set of subgroups of G and Σ be the set of intermediate rings between L(P, F) and F, where L(P, F) is the complete ring of linear transformations of left vector space P F over P. Then (i) every intermediate ring R Σ is a skew group ring H F = L(Inv H, F) for some H Λ (ii) the mappings H H F, R G R, H Λ, R Σ, are inverse mappings and so they are biections of Λ onto Σ and of Σ onto Λ. (iii) a) H 1 H 2 H 1 F H 2 F b) dim F R = G R, [G : G R] L = dim F L(P,F ) dim F for each R Σ. R Proof: (i) By assumption, F R L(P, F). Clearly, ur = F for each 0 u F, that is, F is an irreducible faithful right module over R. By Schur s Lemma, the centralizer K of R on F R is a skew field. It is easy to see that P K F. Let L(K, F) be the complete ring of K-linear transformation of left vector space K F over K. Then R = L(K, F) since K F is finite dimensional and L(P, F) L(K, F) F. Applying Lemma 3.5, there exists a subgroup H = L(K, F) G of G such that R = L(K, F) =H F for H Λ is a skew group subring of L(P, F). (ii) if ψ G HF with ψ ψ H ψ F, then by Lemma 3.3, ψ H. Hence, H H F (H F ) G = H. On the other hand, if R Σ then R = L(K, F) =(G R)F, by Lemma 3.5, where G R is a subgroup of G, that is, R R G (R G)F = R. This completes the proof of (ii). (iii) a) Clearly, by (ii), we have H 1 H 2 H 1 F H 2 F. b) Since R = L(K, F) =(G R)F = ψ G Rψ F is a vector space over F, it is clear that dim F R = G R. Since G = H [G : H], dim F L(P, F) =

62 Xu Yonghua and K. P. Shum (dim F R)[G : H], where R = H F, and L(P, F) =GF, by Lemma 3.3. Thus the proof is completed. Acknowledgement. The authors would like to thank the referee for pointing to them many useful information in the literature about the known results on Galois theory for skew fields. References [1] P. M. Cohn, Skew fields, theory of general division rings, Encyclopedia of Mathematics and its Applications, 57, Cambridge University Press, Cambridge, 1995. [2] P. M. Cohn, On a theorem of skew field construction, Proc. Amer. Math. Soc., 84 (1982), No.1, 1-7. [3] P. M. Cohn, Skew field constructions, Lecture Notes Series, N 27, London Math. Soc., Cambridge University Press, 1977. [4] N. Jacobson, Structure of rings,structure of Rings, American Math.Soc. Colloquim Publications, Vol 27, 1956. [5] N. Jacobson, Basic Algebra 1, W.H. Freeman and Company,1974. [6] E. R. Kolchin, On the Galois theory of differential fields, Amer. J. Math. 77 (1955),868-894. [7] E. R. Kolchin, Galois theory of differential fields, Amer. J. Math.,75 (1953),753-824. [8] Y. H. Xu, A finite strucutre theorem between primitive rings and its applications to Galois theory, Chinese Annals of Math.,Vol 1,( 1980), 183-196. [9] Y. H. Xu, A theory of rings that are isomorphic to the complete rings of linear transformations (vi), Acta Math Sinica, 23 (1980),646-657. Received: September 28, 2007