Let D be a division ring finite dimensional over its center F. The goal of
this paper is to prove that for any positive integer n there exists a in D^(n);
the n-th multiplicative derived subgroup, such that F(a) is a maximal subfield
of D. We also show that a single depth-n iterated additive commutator would
generate a maximal subfield of D.Comment: We have made some editorial changes as well as changes in Lemma 9 and
  Theorem 1

Aaghabali, Mehdi

Bien, Mai Hoang

English

arXiv

summary:Let $D$ be a division ring finite dimensional over its center $F$. The goal of this paper is to prove that for any positive integer $n$ there exists $a\in D^{(n)},$ the $n$th multiplicative derived subgroup such that $F(a)$ is a maximal subfield of $D$. We also show that a single depth-$n$ iterated additive commutator would generate a maximal subfield of $D.

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Certain simple maximal subfields in division rings

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Czechoslovak Mathematical JournalMehdi Aaghabali; Mai Hoang BienCertain simple maximal subfields in division ringsCzechoslovak Mathematical Journal, Vol. 69 (2019), No. 4, 1053–1060Persistent URL: http://dml.cz/dmlcz/147913Terms of use:© Institute of Mathematics AS CR, 2019Institute of Mathematics of the Czech Academy of Sciences provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 69 (144) (2019), 1053–1060CERTAIN SIMPLE MAXIMAL SUBFIELDS IN DIVISION RINGSMehdi Aaghabali, Edinburgh, Tehran, Mai Hoang Bien, Ho Chi Minh CityReceived January 22, 2018. Published online June 5, 2019.Abstract. Let D be a division ring finite dimensional over its center F . The goal of thispaper is to prove that for any positive integer n there exists a ∈ D(n), the nth multiplicativederived subgroup such that F (a) is a maximal subfield of D. We also show that a singledepth-n iterated additive commutator would generate a maximal subfield of D.Keywords: division ring; rational identity; maximal subfieldMSC 2010 : 16K20, 16R50, 17A351. PreliminaryThroughout this paper, D is a division ring with center F. An element a ∈ D iscalled algebraic over F if there exists a nonzero polynomial a0 + a1x + . . . + anxnover F such that a0 + a1a+ . . .+ anan = 0. If a ∈ D, then F (a) denotes the subfieldof D generated by F and {a}. For a (multiplicative) group G we denote by (a, b) =aba−1b−1 the multiplicative commutator of a, b ∈ G and (G,G) the multiplicativecommutator subgroup of G. We denote by G ⊇ G′ ⊇ . . . ⊇ G(n) ⊇ . . . the derivedseries of G, that is, G′ = (G,G) and G(n+1) = (G(n), G(n)) for every n > 1. Fora unital associative ring R we use [a, b] = ab− ba to denote the additive commutatorof a, b ∈ R and R1 = [R,R] the additive commutator subgroup of R. We denoteby R ⊇ R1 ⊇ . . . ⊇ Rn ⊇ . . . the additive derived series of R, that is, R1 = [R,R]and Rn+1 = [Rn, Rn] for every n > 1. For a given division ring D we call D(n)and Dn the nth multiplicative and additive derived groups of D, respectively. In thecase of division ring D we simply use D′ and [D,D] to denote the multiplicative andadditive group of commutators in D, respectively. If A is a subset of D, we use A∗The research of the first author was supported by ERC grant number 320974. Thesecond author was funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant no. C2018-18-03.DOI: 10.21136/CMJ.2019.0039-18 1053to denote A \ {0}. A subfield K of D is called a maximal subfield if K is its owncentralizer in D∗.We denote by dimF D the dimension of D over F. If dimF D = n2,then n is called the degree of division ring D. By Mn(K), GLn(K) and SLn(K) wemean all square matrices, all invertible matrices and all matrices of determinant oneof order n with entries from K, respectively.Mahdavi-Hezavehi in [10] investigates the algebraic properties of the multiplicativegroup of commutators in a division ring and shows that any subfield K of a divisionring D which is separable over the center of D is generated over the center bya commutator subgroup of D′. Afterwards, Mahdavi-Hezavehi and his colleaguesin [12] studied other generating properties of commutator subgroup and showed thateach finite separable extension of the center of D could be considered as a simpleextension F (c), where c is an element in D′. Now, it is natural to consider similarquestions in terms of some other elements coming from certain substructures ofa division ring. In particular, one can pose the following questions:Question 1.1 ([11], Problems 28, 29). Let D be a division ring finite dimensionalover its center F.(i) For any noncentral normal subgroupN ofD∗, does there exist an element c ∈ Nsuch that F (c) is a maximal subfield in D?(ii) For any noncentral subnormal subgroup N of D∗, does there exist an elementc ∈ N such that F (c) is a maximal subfield in D?In this note we rely on rational identities to show that some maximal subfields aregenerated by elements coming from D(n) and Dn, or the nth derived subgroup of D∗and the nth iterated group of additive commutators, for any positive integer n. Thesefall under a wider class of problems concerning the question of whether a noncentralsubnormal subgroup of D∗ cannot be “too small”, and questions about the imagesof (noncommutative) polynomials evaluated on central simple algebras. For n = 1,both results have been proved by Chebotar et al. in [6], Theorem 3, Theorem 6, andrecently again by the authors and Akbari in [1], Theorem 6, Theorem 7. Both [1], [6]and the current paper use rational polynomial identities for proving the aforemen-tioned results. The idea is simple and clever: The key is a certain (noncommutative)polynomial gn(x, y1, . . . , yn) that vanishes whenever an algebraic element of degreenot more than n is substituted into x. One takes n < degD, substitutes a relevantrational expression into x and proves that the resulting expression cannot vanish onD ⊗F L, where L is some splitting field of D. In [1] and [6], the expressions sub-stituted into x are single additive, or multiplicative commutators on two variables,whereas here, iterated commutators are considered.10542. Rational identitiesLet F be a field and X = {x1, . . . , xm} be m noncommuting indeterminates.Denote by F 〈X〉 and F (X), respectively, the free algebra in X over F and theuniversal division ring of fractions of F 〈X〉. A rational expression over F is anelement of F (X). Let R be an F -algebra. A rational expression f over F is said tobe a rational identity of R if it vanishes on all permissible substitutions from R. Inthis case, we say that R satisfies the rational identity f = 0.Example 2.1.(1) It is not hard to see that (Hua’s identity) (x−1+(y−1−x−1)−1)−1−x+xyx = 0is a rational identity of every algebra over an arbitrary field F .(2) One can easily verify that (x + y)−1 − y−1(x−1 + y−1)−1x−1 = 0 is a rationalidentity of every algebra over an arbitrary field F .(3) It is easy to check that ((x, (y, z)x(y, x)−1)3, z) = 0 vanishes on permissiblesubstitutions of M3(F ) for any field F .A rational identity f of an algebra R is called nontrivial if f is nonzero in F (X),see [6]. In the special case when R = D is a division ring, we have some furtherinformation: assume that f = 0 is a rational identity of D. Then f is nontrivialif and only if there exists a division ring L containing all coefficients of f and fis not a rational identity of L. One direction of the statement is trivial, to seethe other direction, assume that f is nontrivial. Then it is well known that thereexists a division ring L with infinite center, which contains F , and that L is infinitedimensional over its center. Hence by [7], f = 0 is not a rational identity of D. Inthe example, it is easily seen that (1) and (2) are trivial, however one can verifythat (3) is nontrivial.In this paper, our algebras R are central simple algebras over a field F . Thatis, R ∼= Mn(D), where D is a division ring which is finite dimensional over F . Wedenote by I(R) the set of all nontrivial rational identities of the algebra R. It isknown that a division ring D with infinite center F is a finite dimensional vectorspace over its center if and only if I(D) 6= ∅, see [7]. Therefore there are rings Rwith I(R) = ∅. Moreover:Theorem 2.2 ([3], Theorem 11). Let F be an infinite field and R be a centralsimple F -algebra with dimF R = n2. Assume that L is a field extension of F . ThenI(R) = I(Mn(F )) = I(Mn(L)).We consider the following example of a rational expression which is important inthis paper. Given a positive integer n and n + 1 noncommutative indeterminates1055x, y1, . . . , yn, putgn(x, y1, . . . , yn) =∑δ∈Sn+1sign(δ)xδ(0)y1xδ(1)y2xδ(2) . . . ynxδ(n),where Sn+1 is the symmetric group of {0, . . . , n} and sign(δ) is the sign of permu-tation δ. This is a rational expression defined in [5] as a mean to test whetheran element is algebraic of degree n. This rational expression may be considered asa generalisation of the characteristic polynomials of matrices of degree n over a field.Lemma 2.3. Let R = Mn(D) be a central simple algebra over its center F . Forany element a ∈ R, the following conditions are equivalent.(1) The element a is algebraic over F of degree less than or equal to n.(2) gn(a, r1, r2, . . . , rn) = 0 for any r1, r2, . . . , rn ∈ R.P r o o f. It is just a corollary of [5], Corollary 2.3.8. 3. Subfields generated by the elements in the nth multiplicativederived subgroupLet n be a positive integer and let x1, . . . , x2n be 2n indeterminates. We will definea special rational polynomial un(x1, . . . , x2n) successively as follows: set u1(x1, x2) =(x1, x2) = x1x2x−11 x−12 and assume that un−1(x1, . . . , x2n−1) is defined. Then we putun(x1, . . . , x2n) = u1(un−1(x1, . . . , x2n−1), un−1(x2n−1+1, . . . , x2n)).This polynomial relates to the solvability of a group: if G is a solvable group oflength 6 n, that is G(n) = 1, then un(a1, . . . , a2n) = 1 for every a1, . . . , a2n ∈ G. Infact, we show the following result.Lemma 3.1. Let un be as above. If G is a group with (multiplicative) derivedseriesG ⊇ G′ ⊇ . . . ⊇ G(n) ⊇ . . . ,then un(a1, . . . , a2n) ∈ G(n) for a1, . . . , a2n ∈ G.P r o o f. We prove the lemma by induction on n. Assume that G is a groupand a1, a2 ∈ G. One has a1a2a−11 a−12 ∈ G′, which implies that u1(a1, a2) ∈ G′.Hence, the lemma holds for u1 and for the group G. Assume that for every group H ,un−1(a1, . . . , a2n−1) ∈ H(n−1) for every a1, . . . , a2n−1 ∈ H . We must prove thatfor every group G, un(b1, . . . , b2n) ∈ G(n) for every b1, . . . , b2n ∈ G. This followsimmediately from the definitions of un and G(n) by induction on n. 1056Lemma 3.2. Let K be an infinite field and m > 1. For any positive integer n andevery nonscalar matrix C ∈ SLm(K) there exist nonscalar matrices A1, . . . , A2n ∈SLm(K) such that C = un(A1, . . . , A2n).P r o o f. We show the lemma by induction on n. Assume that n = 1. Itis well-known that every nonscalar matrix in SLm(K) is a single commutator,see [13]. Hence, there exist nonscalar matrices A1, A2 ∈ SLm(K) such thatC = A1A2A−11 A−12 . Thus, the statement holds for n = 1. Assume that the statementis true for n−1, that is, for every nonscalar matrix C ∈ SLm(K) there exist nonscalarmatrices A1, . . . , A2n−1 ∈ SLm(K) such that C = un−1(A1, . . . , A2n−1). Now by theinduction hypothesis for every nonscalar matrix C there exist nonscalar matricesB1, B2, A1, . . . , A2n ∈ SLm(K) such that C = u1(B1, B2), B1 = un−1(A1, . . . , A2n−1)and B2 = un−1(A2n−1+1, . . . , A2n). ThereforeC = u1(B1, B2) = u1(un−1(A1, . . . , A2n−1), un−1(A2n−1+1, . . . , A2n))= un(A1, . . . , A2n).This implies that the statement is true for n. Before showing the main result of this section, we recall the following well-knownlemma.Lemma 3.3 ([9], page 242). Let D be a division ring with center F and K bea subfield of D containing F . If dimF D = m2, then dimF K 6 m. The equalityholds if and only if K is a maximal subfield of D.Theorem 3.4. Let D be a division ring finite dimensional over a its center F. Forany positive integer n there exists a ∈ D(n), the nth multiplicative derived subgroup,such that F (a) is a maximal subfield of D.P r o o f. If F is finite, thenD is also finite and we have nothing to prove. Supposethat F is infinite and dimF D = m2. By Lemma 3.3, it suffices to show that thereexists a ∈ D(n) such that dimF F (a) > m. Indeed, putl = max{dimF F (un(a1, . . . , a2n)) : a1, . . . , a2n ∈ D∗}.Applying Lemma 2.3 we see that gl(un(a1, . . . , a2n), r1, . . . , rl) = 0 for any r1, . . . ,rl ∈ D and a1, . . . , a2n ∈ D∗. In other words,gl(un(x1, . . . , x2n), y1, . . . , yl) = 01057is a rational identity of D. It is not hard to verify that gl(un(x1, . . . , x2n), y1, . . . , yl)is a nonzero element of F (x1, . . . , x2n , y1, . . . , yl) (see [8], Theorem 3.4). Hence, byTheorem 2.2 it is also a rational identity of Mn(F ). This yields thatgl(un(A1, . . . , A2n), B1, . . . , Bl) = 0for all matrices Ai ∈ GLm(F ) and Bi ∈ Mm(F ). In view of Lemma 2.3, un(A1, . . . ,A2n) is algebraic over F of degree not more than l for every A1, . . . , A2n ∈ Mm(F ).Now consider the (m×m)-matrix T = (tij)16i,j6m as follows: if j = i or j = i+ 1,then tij = 1, otherwise tij = 0. It is easy to see that T ∈ SLm(F ) and T is algebraicof degree m over F . By Lemma 3.2, one can find matrices A1, . . . , A2n ∈ SLm(F )such that un(A1, . . . , A2n) = T . Hence, l > m. This completes the proof. 4. Subfields generated by the elements in the nth additivederived subgroupLet n be a positive integer and let x1, . . . , x2n be 2n indeterminates. We de-fine a polynomial vn(x1, . . . , x2n) successively as follows: set v1(x1, x2) = [x1, x2] =x1x2 − x2x1. Assume that vn−1(x1, . . . , x2n−1) is defined. Then we putvn(x1, . . . , x2n) = v1(vn−1(x1, . . . , x2n−1), vn−1(x2n−1+1, . . . , x2n)).Lemma 4.1. Let R be an algebra with additive derived seriesR ⊇ R1 ⊇ . . . ⊇ Rn ⊇ . . . .If vn is defined as above, then vn(a1, . . . , a2n) ∈ Rn for a1, . . . , a2n ∈ R.P r o o f. The proof is similar to that of Lemma 3.1. Lemma 4.2. Let K be a field and m > 1 such that charK ∤ m. For any pos-itive integer n and every matrix C ∈ Mm(K) with zero-trace there exist matricesA1, . . . , A2n ∈ Mm(K) whose trace is zero and C = vn(A1, . . . , A2n).P r o o f. The idea of the proof is similar to that of Lemma 3.2. We prove thelemma by induction on n. Assume that n = 1. In view of [2], which states thatevery matrix in Mn(K) with zero-trace is a single additive commutator, there existA1, A2 ∈ Mm(K) such that C = A1A2 − A2A1. Set B1 = A1 − (tr(A1)/m)Imand B2 = A2 − (tr(A2)/m)Im, where by tr(A) we mean the trace of A. We haveC = B1B2 − B2B1 = v1(B1, B2) and tr(B1) = tr(B2) = 0. Hence, the statementholds for n = 1. The general case follows by induction on n, similarly to the proofof Lemma 3.2. 1058The following result is the goal of this section. Note that the same result as The-orem 3.4, however, follows from the case where n = 1, proved in [1] and [6], since bya theorem of Amitsur and Rowen, D1 = D2 = D3 = . . ., see [4]. However, the proofyields a slightly stronger claim that a single depth-n iterated additive commutatorwould generate a maximal subfield, which does not follow from D1 = D2 = D3 = . . .Theorem 4.3. Let D be a division ring finite dimensional over its center F ofcharacteristic either zero or a prime p such that p ∤ dimF D. For any positive integer nthere exists a depth-n iterated additive commutator which generates a maximalsubfield of D.P r o o f. First note that if charF = 0, then by a result due to Amitsur andRowen in [4] we have D1 = D2 = . . . Hence, in this case the result follows from [1],Theorem 7. In case of charF = p > 0, the proof is similar to the one of Theorem 3.4.We assume that F is infinite since if F is finite, then D is also finite and there isnothing to prove. Suppose dimF D = m2. In view of Lemma 3.3, it suffices to showthat there exists a ∈ Dn such that dimF F (a) > m. Indeed, putl = max{dimF F (vn(a1, . . . , a2n)) : a1, . . . , a2n ∈ D∗}.According to Lemma 2.3 we see that gl(vn(a1, . . . , a2n), r1, . . . , rl) = 0 for anyr1, . . . , rl ∈ D and a1, . . . , a2n ∈ D∗. In other words,gl(vn(x1, . . . , x2n), y1, . . . , yl) = 0is a polynomial identity of D, so it is also a rational identity ofMn(F ) (Lemma 2.2).Note that it is easily seen that gl(vn(x1, . . . , x2n), y1, . . . , yl) is a nonzero element ofF (x1, . . . , x2n , y1, . . . , yl) (see [8], Theorem 3.4). This yields thatgl(vn(A1, . . . , A2n), B1, . . . , Bl) = 0for all matrices Ai, Bi ∈ Mm(F ). According to Lemma 2.3, vn(A1, . . . , A2n) is alge-braic over F of degree not more than l for every A1, . . . , A2n ∈ Mm(F ). Now considerthe (m×m)-matrix T = (tij)16i,j6m defined by ti(i+1) = 1 and tij = 0 if j 6= i+ 1.We can show that tr(T ) = 0 and T is algebraic of degree m over F . By Lemma 4.2,there exist matrices A1, . . . , A2n ∈ Mm(F ) such that vn(A1, . . . , A2n) = T . Hence,l > m and this completes the proof. 1059References[1] M.Aaghabali, S. Akbari, M.H.Bien: Division algebras with left algebraic commutators.Algebr. Represent. Theory 21 (2018), 807–816. zbl MR doi[2] A.A.Albert, B.Muckenhoupt: On matrices of trace zeros. Mich. Math. J. 4 (1957), 1–3. zbl MR doi[3] S.A.Amitsur: Rational identities and applications to algebra and geometry. J. Algebra3 (1966), 304–359. zbl MR doi[4] S.A.Amitsur, L. H.Rowen: Elements of reduced trace 0. Isr. J. Math. 87 (1994),161–179. zbl MR doi[5] K. I.Beidar, W. S.Martindale, III, A.V.Mikhalev: Rings with Generalized Identities.Pure and Applied Mathematics 196, Marcel Dekker, New York, 1996. zbl MR[6] M.A.Chebotar, Y. Fong, P.-H. Lee: On division rings with algebraic commutators ofbounded degree. Manuscr. Math. 113 (2004), 153–164. zbl MR doi[7] K.Chiba: Generalized rational identities of subnormal subgroups of skew fields. Proc.Am. Math. Soc. 124 (1996), 1649–1653. zbl MR doi[8] B.X.Hai, T. H.Dung, M.H. Bien: Almost subnormal subgroups in division ringswith generalized algebraic rational identities. Available at https://arxiv.org/abs/1709.04774.[9] T.Y. Lam: A First Course in Noncommutative Rings. Graduate Texts in Mathematics131, Springer, New York, 2001. zbl MR doi[10] M.Mahdavi-Hezavehi: Extension of valuations on derived groups of division rings. Com-mun. Algebra 23 (1995), 913–926. zbl MR doi[11] M.Mahdavi-Hezavehi: Commutators in division rings revisited. Bull. Iran. Math. Soc.26 (2000), 7–88. zbl MR[12] M.Mahdavi-Hezavehi, S. Akbari-Feyzaabaadi, M.Mehraabaadi, H.Hajie-Abolhassan:On derived groups of division rings. II. Commun. Algebra 23 (1995), 2881–2887. zbl MR doi[13] R.C.Thompson: Commutators in the special and general linear groups. Trans. Am.Math. Soc. 101 (1961), 16–33. zbl MR doiAuthors’ addresses: M e h d i A a g h a b a l i (corresponding author), School of Mathe-matics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland; School of Mathematics,Statistics and Computer Science, University of Tehran, Tehran, Iran, e-mail: maghabali@gmail.com; M a i H o a n g B i e n, Faculty of Mathematics and Computer Science, Univer-sity of Science, VNU-HCM, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam,e-mail: mhbien@hcmus.edu.vn.1060

Certain Simple Maximal Subfields in Division Rings

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