On a class of topological groups. Key Words: topological groups, linearly topologized groups. Contents. 1 Introduction 25

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1 Bol. Soc. Paran. Mat. (3s.) v (2006): c SPM ISNN On a class of topological grops Dinamérico P. Pombo Jr. abstract: A topological grop is an SNS-grop if its identity element possesses a fndamental system of neighborhoods formed by normal sbgrops. In this paper we prove the existence of initial SNS-topologies, from which we derive that the class of SNS-grops is closed nder the formation of prodcts and projective limits, and we prove the existence of final SNS-topologies, from which we derive that the class of SNS-grops is closed nder the formation of free prodcts and indctive limits. Key Words: topological grops, linearly topologized grops. Contents 1 Introdction 25 2 Initial and final SNS-topologies Introdction Althogh the class of linearly topologized grops (a linearly topologized grop is an abelian topological grop whose identity element possesses a fndamental system of neighborhoods formed by sbgrops) is qite rich in examples, there exist important non-abelian SNS-grops which occr, for example, in Galois Theory. This fact has motivated s to write the present paper, where basic reslts concerning SNS-grops have been established. The main prpose of the paper is the discssion of the fndamental constrctions in the class of SNS-grops. In this regard, it is shown that this class is closed nder the formation of prodcts, projective limits, free prodcts and indctive limits. Moreover, we emphasize the niversal properties satisfied by the objects which have been constrcted. It shold also be mentioned that, althogh part of the reslts presented here are well-known in the mathematical folklore, the ones concerning topological free prodcts and topological indctive limits are possibly new. Throghot this paper G is an arbitrary grop whose identity element is denoted by e, nless otherwise specified. In the whole paper, the operation on a grop has been written mltiplicatively. Definition 1. A topological grop (G, τ) is said to be an SNS-grop, and τ is said to be an SNS-topology on G, if e admits a fndamental system of τ-neighborhoods consisting of normal sbgrops of G Mathematics Sbject Classification: 22A30 Date sbmission 03-Jl Typeset by B S P M style. c Soc. Paran. Mat.

2 26 Dinamérico P. Pombo Jr. In the case where G is an abelian grop, to say that τ is an SNS-topology on G is eqivalent to saying that τ is a linear topology on G; we refer to [5] and [6] for the stdy of sch topologies. It is easily seen that the bilateral completion ([9], 5) of a separated SNS-grop is an SNS-grop. Example 2. The discrete topology and the chaotic topology on a grop G are SNS-topologies on G. Example 3. If (E, τ) is a linearly topologized modle ([9], Definition 31.4), then the nderlying additive topological grop is an SNS-grop. Example 4. If (E, τ) is a non-archimedean topological vector space over a nontrivially valed division ring ([7], Definition 3.2), then the nderlying additive topological grop is an SNS-grop in view of Theorem 3.7 of [7]. Example 5. Let A be a non-empty set and (G, τ) an SNS-grop. If F(A, G) is the grop of all mappings from A into G and τ A is the topology of niform convergence on F(A, G), then (F(A, G), τ A ) is an SNS-grop. In fact, it is easily seen that τ A is a grop topology on F(A, G). Moreover, if V is a fndamental system of τ-neighborhoods of e consisting of normal sbgrops of G, then the sets {f F(A, G); f(a) V } (V V) form a fndamental system of τ A -neighborhoods of the identity element of F(A, G) consisting of normal sbgrops of F(A, G). Proposition 6. If B is a filter base on a grop G consisting of normal sbgrops of G, then there exists a niqe SNS-topology on G for which B is a fndamental system of neighborhoods of e. Proof. Follows immediately from Corollary 1.5 of [9]. Example 7. Let G be an abelian grop and let p be a positive prime nmber. It is easily seen that B = {p n G; n N} is a filter base on G consisting of sbgrops of G (observe that p m G p n E = p n G if n m). By Proposition 6, there exists a niqe SNS-topology on G for which B is a fndamental system of neighborhoods of e, called the p-adic topology on G. Example 8. Let F, G and H be three abelian grops and let e F (resp. e G, e H ) be the identity element of F (resp. G, H). Let B : F G H be a Z-bilinear mapping. For each finite sbset {y 1,..., y m } of G pt U {y1,...,y m } = {x F ; B(x, y i) = e H for i = 1,..., m}. It is easily seen that the set B formed by all the sets U {y1,...,y m} is a filter base on F consisting of sbgrops of F. By Proposition 6, there exists a niqe SNS-topology on F for which B is a fndamental system of neighborhoods of e F. Similarly, for each finite sbset {x 1,..., x m } of F pt V {x1,...,x m} = {y G; B(x i, y) = e H for i = 1,..., m}.

3 On a class of topological grops 27 Then all the sets V {x1,...,x m} constitte a fndamental system of neighborhoods of e G for a niqe SNS-topology on G. Let s mention two basic examples of Z-bilinear mappings. For F, G and H as above, let Hom(F, G) (resp. Hom(G, H), Hom(F, H)) be the abelian grop of all grop homomorphisms from F into G (resp. G into H, F into H). Then the mappings (x, ) F Hom(F, G) (x) G and (, v) Hom(F, G) Hom(G, H) v Hom(F, H) are Z-bilinear. Example 9. Let N be a Galois extension of a field K and Γ the Galois grop of N over K. For each intermediate field K L N which is a finite Galois extension of K, let g(l) be the Galois grop of N over L; then g(l) is a normal sbgrop of Γ and the set B formed by all the sets g(l) constittes a filter base on Γ (see Appendix II of [3] for the details). By Proposition 6, there exists a niqe SNS-topology on Γ for which B is a fndamental system of neighborhoods of the identity element 1 N of Γ. 2. Initial and final SNS-topologies Now let s begin the discssion of initial SNS-topologies. Theorem 10. Let ((G i, τ i )) be a non-empty family of SNS-grops, G a grop and, for each i I, let i : G G i be a grop homomorphism. If τ is the initial topology on G for the family ((G i, τ i ), i ) ([2], p.28, Proposition 4), then (G, τ) is an SNS-grop. Proof. By Theorem 1.9 of [9], (G, τ) is a topological grop. For each i I let V i be a fndamental system of τ i -neighborhoods of the identity element e i of G i consisting of normal sbgrops of G i. For each finite sbset {i 1,..., i m } of I and for each V i1 V i1,..., V im V im, consider the set 1 i 1 (V i1 ) 1 i m (V im ), which is a normal sbgrop of G. Since all these sets constitte a fndamental system of τ-neighborhoods of e, then (G, τ) is an SNS-grop, as was to be shown. Remark 11. Under the conditions of Theorem 10, if τ i is the chaotic topology on G i for all i I, then τ is the chaotic topology on G. Example 12. Let X be a non-empty set and A a set of non-empty sbsets of X. Let (G, τ) be an SNS-grop and F(X, G) the grop of all mappings from X into G. For each A A consider the grop homomorphism A : f F(X, G) f A F(A, G). By Theorem 10, F(X, G) endowed with the initial topology τ A for the family ((F(A, G), τ A ), A ) A A is an SNS-grop ((F(A, G), τ A ) being as in Example 5). τ A is the topology of A-convergence on F(X, G). Corollary 13. Let (H, θ) be an SNS-grop, G a grop and : G H a grop homomorphism. If τ is the inverse image of θ nder, then (G, τ) is an SNS-grop.

4 28 Dinamérico P. Pombo Jr. In particlar, every sbgrop of an SNS-grop is an SNS-grop nder the indced topology. Proof. Follows immediately from Theorem 10. Example 14. Let (X, τ) be a non-empty topological space, A a set of non-empty sbsets of X, (G, θ) an SNS-grop and C(X, G) the sbgrop of F(X, G) consisting of all continos mappings from (X, τ) into (G, θ). Then, in view of Corollary 13, C(X, G) is an SNS-grop nder the topology indced by τ A (τ A being as in Example 12). Corollary 15. If ((G i, τ i )) is a non-empty family of SNS-grops, G is the prodct grop G i and τ is the prodct topology τ i on G, then (G, τ) is an SNS-grop. Proof. Follows immediately from Theorem 10. Remark 16. Let ((G i, τ i )) and (G, τ) be as in Corollary 15 and, for each i I, let pr i : G G i be the projection on the i-th factor. Then (G, τ) satisfies the following niversal property: for each SNS-grop (H, θ), the mapping Hom c (H, G) (pr i ) Hom c (H, G i ) is a grop isomorphism, where Hom c (H, G) is the grop of all continos grop homomorphisms from (H, θ) into (G, τ), Hom c (H, G i ) is the grop of all continos grop homomorphisms from (H, θ) into (G i, τ i ) for all i I and Hom c (H, G i ) is the corresponding prodct grop. Corollary 17. If (τ i ) is a non-empty family of SNS-topologies on a grop G and τ = sp τ i, then (G, τ) is an SNS-grop. Proof. Follows immediately from Theorem 10. Definition 18. Let ((G i, τ i ), ij ) be a projective system of SNS-grops (this means that I is a non-empty set endowed with a partial order, (G i, τ i ) is an SNSgrop for all i I, ij : (G j, τ j ) (G i, τ i ) is a continos grop homomorphism for i, j I with i j, ii = 1 Gi for all i I and ik = ij jk for i, j, k I with i j k). (G k, τ k ) ik (Gi, τ i ) jk ij (G j, τ j ) For each i I let pr i : G = G i G i be the projection on the i-factor. Pt lim G i = {x G; ( ij pr j )(x) = pr i (x) for all i, j I with i j}, which is a sbgrop of the prodct grop G = G i (note that lim G i = G if is the eqality relation). For each i I let i = pr i (lim G i ), which is called the canonical grop homomorphism from lim G i into G i. If τ is the initial topology on lim G i for the family ((G i, τ i ), i ), which makes lim G i an SNS-grop in view

5 On a class of topological grops 29 of Theorem 10, then (lim G i, τ) is said to be the topological projective limit of the system ((G i, τ i ), ij ). Remark 19. The projective limit topology for the system ((G i, τ i ), ij ) is an SNS-topology since, by definition ([2], p.51), it is precisely the topology τ considered in Definition 18. Proposition 20. Let (lim G i, τ) be the topological projective limit of the projective system ((G i, τ i ), ij ) of SNS-grops. Then the topology θ on lim G i indced by the prodct topology τ i coincides with τ. Proof. Let H = lim G i, and let pr i and i be as in Definition 18. For each i I let V i be the set of all τ i -neighborhoods of the identity element e i of G i. If i 1,..., i m I, V i1 V i1,..., V im V im, we have [ pr 1 i 1 (V i1 ) pr 1 i m (V im ) ] H = ( pr 1 i 1 (V i1 ) H ) ( pr 1 i m (V im ) H ) = 1 i 1 (V i1 ) 1 i m (V im ). Conseqently, θ = τ, as asserted. Example 21. Let p be a positive prime nmber. Let Z be the additive grop of integers and, for each positive integer m, let G m be the qotient grop Z / p m Z endowed with the discrete topology τ m ((G m, τ m ) is an SNS-grop by Example 2). For m n let mn : G n G m be the canonical grop homomorphism, which is obviosly continos from (G n, τ n ) into (G m, τ m ). Then ((G m, τ m ), mn ) m N is a projective system of compact SNS-grops, and hence we can consider the topological projective limit (lim G m, τ) of this system. Since lim G m is closed in ( ) ( ) G m, τ m and G m, τ m is compact, it follows from Proposition 20 that (lim G m, τ) is compact. If Z p is the linearly topologized ring of p-adic m N m N m N m N integers ([8], p.11), then the nderlying additive topological grop is (lim G m, τ). The topological projective limit (lim G i, τ) of the projective system ((G i, τ i ), ij ) of SNS-grops satisfies the following niversal property: Proposition 22. Let (H, θ) be an SNS-grop and, for each i I, let α i : (H, θ) (G i, τ i ) be a continos grop homomorphism sch that ij α j = α i for i j. Then there exists a niqe continos grop homomorphism : (H, θ) (lim G i, τ) sch that α i = i for all i I ( i being as in Definition 18). (G j, τ j ) α j ij (Gi, τ i ) α i (H, θ) (H, θ) α i (G i, τ i ) (lim G i, τ) i n Proof. Let y H be arbitrary. It is clear that (y) = (α i (y)) is the niqe element of G i satisfying α i (y) = pr i ((y)) for all i I. Moreover, (y) lim G i becase ( ij α j )(y) = α i (y) for i j. It then follows that the mapping : H lim G i so defined is the niqe grop homomorphism sch that α i = i for all

6 30 Dinamérico P. Pombo Jr. i I. Finally, : (H, θ) (lim G i, τ) is continos becase i : (H, θ) (G i, τ i ) is continos for all i I. Corollary 23. Let ((G i, τ i ), ij ) and ((H i, θ i ), v ij ) be two projective systems of SNS-grops, and let (lim G i, τ) and (lim H i, θ) be the corresponding topological projective limits. For each i I let β i : (G i, τ i ) (H i, θ i ) be a continos grop homomorphism sch that v ij β j = β i ij for i j. Then there exists a niqe continos grop homomorphism : (lim G i, τ) (lim H i, θ) sch that v i = β i i for all i I, where i : lim G i G i and v i : lim H i H i are the canonical grop homomorphisms (i I). β j (G j, τ j ) (Hj, θ j ) ij v ij (G i, τ i ) (H i, θ i ) βi (lim G i, τ) (lim H i, θ) i v i (G i, τ i ) (H i, θ i ) βi Proof. Pt α i = β i i for i I; then α i is a continos grop homomorphism from (lim G i, τ) into (H i, θ i ). Since v ij α j = (v ij β j ) j = (β i ij ) j = β i ( ij j ) = β i i = α i for i j, Proposition 22 garantees the existence of a niqe continos grop homomorphism : (lim G i, τ) (lim H i, θ) sch that α i = v i for all i i. This completes the proof. Now let s trn to the discssion of final SNS-topologies. Theorem 24. Let ((G i, τ i )) be a non-empty family of SNS-grops and let G be a grop. For each i I let i : G i G be a grop homomorphism. Then there exists a niqe SNS-topology τ on G which is final for the family ((G i, τ i ), i ), in the following sense: for every SNS-grop (H, θ) and for every grop homomorphism : G H, we have that : (G, τ) (H, θ) is continos if and only if i : (G i, τ i ) (H, θ) is continos for all i I. Proof. For each i I let V i be the set of all τ i -neighborhoods of the identity element e i of G i. Pt B = {U G; U is a normal sbgrop of G and 1 i (U) V i for all i I}. Clearly, B is a filter base on G. Let τ be the SNS-topology on G for which B is a fndamental system of τ-neighborhoods of e (Proposition 6). By constrction, i : (G i, τ i ) (G, τ) is continos for all i I. We claim that τ is final for the family ((G i, τ i ), i ). Indeed, let (H, θ) be an SNS-grop and let : G H be a grop homomorphism. If : (G, τ) (H, θ) is continos, then i : (G i, τ i ) (H, θ) is continos for all i I. Conversely, assme that i : (G i, τ i ) (H, θ) is continos for all i I, and let V be a θ-neighborhood of the identity element f of H which is a normal sbgrop of H. Then 1 (V ) is a normal sbgrop of G and 1 i ( 1 (V )) = ( i ) 1 (V ) V i for all i I; ths 1 (V ) B. Therefore : (G, τ) (H, θ) is continos, proving or claim.

7 On a class of topological grops 31 In order to prove the niqeness, let τ be an SNS-topology on G sch that i : (G i, τ i ) (G, τ) is continos for all i I, and consider the identity mapping 1 G : (G, τ) (G, τ). Since 1 G i : (G i, τ i ) (G, τ) is continos for all i I, it follows that 1 G : (G, τ) (G, τ) is continos. Ths τ is the finest SNS-topology on G which makes all the i continos, and hence the niqeness is established. This completes the proof. Remark 25. Under the conditions of Theorem 24, if τ i is the discrete topology on G i for all i I, then τ is the discrete topology on G. Corollary 26. Every non-empty family of SNS-topologies on a grop G admits an infimm in the partially ordered set of all SNS-topologies on G. Proof. Follows immediately from Theorem 24. Corollary 27. Let (G, τ) be an SNS-grop, H a normal sbgrop of G and π : G G/H the canonical srjection. Then the qotient topology τ on G/H coincides with the final SNS-topology τ for the pair ((G, τ), π). Proof. It is easily seen that (G/H, τ ) is an SNS-grop and that τ is finer than τ. Therefore τ = τ, as asserted. Definition 28. Let ((G i, τ i )) be a non-empty family of SNS-grops and let G be the free prodct of the family (G i ) ([1]; [4], p.24). For each i I let i : G i G be the canonical grop homomorphism. If τ is the final SNS-topology on G for the family ((G i, τ i ), i ), (G, τ) is said to be the topological free prodct of the family ((G i, τ i )). Remark 29. Let ((G i, τ i )), G, i and τ be as in Definition 28. Then (G, τ) satisfies the following niversal property: for each SNS-grop (H, θ), the mapping Hom c (G, H) ( i ) Hom c (G i, H) is a grop isomorphism, where Hom c (G, H) is the grop of all continos grop homomorphisms from (G, τ) into (H, θ), Hom c (G i, H) is the grop of all continos grop homomorphisms from (G i, τ i ) into (H, θ) for all i I and Hom c (G i, H) is the corresponding prodct grop. Before we proceed let s establish an axiliary reslt which is probably known: Proposition 30. Let T be a non-empty sbset of a grop G. Then the normal sbgrop of G generated by T (that is, the smallest normal sbgrop of G containing T ) is the set N of all finite prodcts of elements of the form g t g 1, where g G and t is either an element of T or the inverse of an element of T. Proof. Obviosly, e N and ab 1 N for all a, b N. Let g G and a = g 1 t 1 g 1 1 g 2 t 2 g g m t m g 1 m N. Then g a g 1 = (gg 1 )t 1 (gg 1 ) 1 (gg 2 )t 2 (gg 2 ) 1... (gg m )t m (gg m ) 1 N. Ths N is a normal sbgrop G, which clearly contains T. Finally, it is clear that any normal sbgrop of G containing T also contains N. Proposition 31. Let (G i, ji ) be an indctive system of grops and let G be the free prodct of the family (G i ). Let T be the sbset of G formed by all elements of the form i (x i )( j ji )(x 1 i ) (i j, x i E i ), i being as in Definition 28, and let N be the normal sbgrop of G generated by T. Let lim G i be the qotient grop G/N, π : G lim G i the canonical srjection and, for each

8 32 Dinamérico P. Pombo Jr. i I, let v i = π i be the canonical grop homomorphism. Then v j ji = v i for i j. Moreover, if H is a grop and, for each i I, α i : G i H is a grop homomorphism sch that α j ji = α i for i j, then there exists a niqe grop homomorphism : lim G i H sch that α i = v i for all i I. G i i v i G lim G i π G i α i ji H Gj α j lim G i v i G i H α i Proof. Firstly, let s verify that v j ji = v i for i j. In fact, for all x i E i, v i (x i ) ( (v j ji ) (x i ) ) 1 = (π i )(x i ) ( (π j ji )(x i ) ) 1 = (π i )(x i ) ( (π j ji )(x 1 i ) ) = π ( i (x i )( j ji )(x 1 i ) ) = N. Now, let ū: G H be the niqe grop homomorphism sch that ū i = α i for all i I. We claim that N Ker(ū). In fact, let f be the identity element of H. For i j and x i E i, we have ū ( i (x i )( j ji )(x 1 i ) ) = (ū i )(x i )ū ( ( j ji )(x 1 i ) ) = α i (x i )(ū j )( ji (x 1 i )) = α i (x i )(α j ji )(x 1 i ) = α i (x i )α i (x 1 i ) = f. Therefore it follows from Proposition 30 that N Ker(ū). By the isomorphism theorem, there exists a niqe grop homomorphism : lim G i H sch that ū = π. Moreover, v i = ( π) i = ū i = α i for all i I. Finally, let v : lim G i H be a grop homomorphism sch that v v i = α i for all i I, and pt v = v π. Then v i = v (π i ) = v v i = α i for all i I. Conseqently, v = ū, and hence v =. This completes the proof. Definition 32. Let ((G i, τ i ), ji ) be an indctive system of SNS-grops (this means that I is a non-empty set endowed with a partial order, (G i, τ i ) is an SNSgrop for all i I, ji : (G i, τ i ) (G j, τ j ) is a continos grop homomorphism for i, j I with i j, ii = 1 Gi for all i I and ki = kj ji for i, j, k I with i j k). Let lim G i and v i (i I) be as in Proposition 31. If τ is the final SNS-topology on lim G i for the family ((G i, τ i ), v i ), then (lim G i, τ) is said to be the topological indctive limit of the system ((G i, τ i ), ji ). Proposition 33. Let (lim G i, τ) be the topological indctive limit of the indctive system ((G i, τ i ), ji ) of SNS-grops and let (G, τ) be the topological free prodct of the family ((G i, τ i )). Then the qotient topology τ on G / N(= lim G i ) coincides with τ, N being as in Proposition 31.

9 On a class of topological grops 33 Proof. Let π, i and v i be as in Proposition 31. Since v i : (G i, τ i ) (lim G i, τ ) is continos for all i I, τ is coarser than τ. Conversely, let U be a normal sbgrop of lim G i sch that v 1 i (U) is a τ i - neighborhood of the identity element of G i for all i I (U is a basic τ-neighborhood of the identity element of lim G i ). Since v i = π i, 1 i (π 1 (U)) is a τ i - neighborhood of the identity element of G i for all i I. Therefore the normal sbgrop π 1 (U) of G is a τ-neighborhood of the identity element of G, and hence U is a τ -neighborhood of the identity element of lim G i. Ths τ is coarser than τ, and the eqality τ = τ is established. The topological indctive limit (lim G i, τ) of the indctive system ((G i, τ i ), ji ) of SNS-grops satisfies the following niversal property: Proposition 34. Let (H, θ) be an SNS-grop and, for each i I, let α i : (G i, τ i ) (H, θ) be a continos grop homomorphism sch that α j ji = α i for i j. Then there exists a niqe continos grop homomorphism : (lim G i, τ) (H, θ) sch that α i = v i for all i I (v i being as in Proposition 31). (G i, τ i ) α i ji (Gj, τ j ) α j (H, θ) (lim G i, τ) v i (H, θ) α i (G i, τ i ) Proof. By Proposition 31, there exists a niqe grop homomorphism : lim G i H sch that α i = v i for all i I. Moreover, since v i : (G i, τ i ) (H, θ) is continos for all i I, then : (lim G i, τ) (H, θ) is continos. This completes the proof. Corollary 35. Let ((G i, τ i ), ji ) and ((H i, θ i ), v ji ) be two indctive systems of SNS-grops, and let (lim G i, τ) and (lim H i, θ) be the corresponding topological indctive limits. For each i I let β i : (G i, τ i ) (H i, θ i ) be a continos grop homomorphism sch that v ji β i = β j ji for i j. Then there exists a niqe continos grop homomorphism : (lim G i, τ) (lim H i, θ) sch that v i = w i β i for all i I, where v i : G i lim G i and w i : H i lim H i are the canonical grop homomorphisms (i I). β i (G i, τ i ) (Hi, θ i ) ji v ji (G j, τ j ) βj (H j, θ j ) (lim G i, τ) (lim H i, θ) v i w i (G i, τ i ) (H i, θ i ) βi Proof. For each i I pt α i = w i β i ; then α i is a continos grop homomorphism from (G i, τ i ) into (lim H i, θ). Since α j ji = w j (β j ji ) = w j (v ji β i ) = (w j v ji ) β i = w i β i = α i for i j, Proposition 34 garantees the existence of a niqe continos grop homomorphism : (lim G i, τ) (lim H i, θ) sch that α i = v i for all i I. This completes the proof.

10 34 Dinamérico P. Pombo Jr. References [1] E. Artin, The free prodct of grops, Amer. J. Math. 69 (1947), 1 4. [2] N. Borbaki, Topologie Générale, Chapitres I et II, Qatrième édition, Actalités Scientifiqes et Indstrielles 1142, Hermann (1965). [3] N. Borbaki, Algèbre, Chapitres IV et V, Dexième édition, Actalités Scientifiqes et Indstrielles 1102, Hermann (1967). [4] P.J. Hilton and Yel-Chiang W, A Corse in Modern Algebra, Wiley - Interscience Series of Texts, Monographs and Tracts in Pre and Applied Mathematics, John Wiley & Sons (1974). [5] M. Janvier, Topologies linéaires sr les gropes abéliens, Pblications de l Université de Montpellier 98 ( ), [6] D.P. Pombo Jr., Linear topologies on grops: basic constrctions, Atas do 53 ō Seminário Brasileiro de Análise (2001), [Page 74, line 8: read E = lim E i, and let in place of E,.] [7] J.B. Prolla, Topics in Fnctional Analysis over Valed Division Rings, Notas de Matemática 89, North-Holland (1982). [8] J.-P. Serre, A Corse in Arithmetic, Gradate Texts in Mathematics 7, Springer-Verlag (1985). [9] S. Warner, Topological Fields, Notas de Matemática 126, North-Holland (1989). Dinamérico P. Pombo Jr. Institto de Matemática, Universidade Federal Flminense Ra Mário Santos Braga, s/n o., Niterói CEP RJ- Brazil address: dpombo@terra.com.br