Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 69, No. 4, pp. 1053-1060, 2019

Certain simple maximal subfields in division rings

Mehdi Aaghabali, Mai Hoang Bien

Received January 22, 2018. Published online June 5, 2019.

Abstract: 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 $D.$

Keywords: division ring; rational identity; maximal subfield

Classification MSC: 16K20, 16R50, 17A35

DOI: 10.21136/CMJ.2019.0039-18

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] M. Aaghabali, S. Akbari, M. H. Bien: Division algebras with left algebraic commutators. Algebr. Represent. Theory 21 (2018), 807-816. DOI 10.1007/s10468-017-9739-3 | MR 3826728 | Zbl 1397.17004
[2] A. A. Albert, B. Muckenhoupt: On matrices of trace zeros. Mich. Math. J. 4 (1957), 1-3. DOI 10.1307/mmj/1028990168 | MR 0083961 | Zbl 0077.24304
[3] S. A. Amitsur: Rational identities and applications to algebra and geometry. J. Algebra 3 (1966), 304-359. DOI 10.1016/0021-8693(66)90004-4 | MR 0191912 | Zbl 0203.04003
[4] S. A. Amitsur, L. H. Rowen: Elements of reduced trace 0. Isr. J. Math. 87 (1994), 161-179. DOI 10.1007/BF02772992 | MR 1286824 | Zbl 0852.16012
[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). MR 1368853 | Zbl 0847.16001
[6] M. A. Chebotar, Y. Fong, P.-H. Lee: On division rings with algebraic commutators of bounded degree. Manuscr. Math. 113 (2004), 153-164. DOI 10.1007/s00229-003-0430-0 | MR 2128544 | Zbl 1054.16012
[7] K. Chiba: Generalized rational identities of subnormal subgroups of skew fields. Proc. Am. Math. Soc. 124 (1996), 1649-1653. DOI 10.1090/S0002-9939-96-03127-9 | MR 1301016 | Zbl 0859.16014
[8] B. X. Hai, T. H. Dung, M. H. Bien: Almost subnormal subgroups in division rings with 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 Mathematics 131, Springer, New York (2001). DOI 10.1007/978-1-4419-8616-0 | MR 1838439 | Zbl 0980.16001
[10] M. Mahdavi-Hezavehi: Extension of valuations on derived groups of division rings. Commun. Algebra 23 (1995), 913-926. DOI 10.1080/00927879508825257 | MR 1316740 | Zbl 0833.16014
[11] M. Mahdavi-Hezavehi: Commutators in division rings revisited. Bull. Iran. Math. Soc. 26 (2000), 7-88. MR 1828953 | Zbl 0983.16012
[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. DOI 10.1080/00927879508825374 | MR 1332151 | Zbl 0866.16012
[13] R. C. Thompson: Commutators in the special and general linear groups. Trans. Am. Math. Soc. 101 (1961), 16-33. DOI 10.1090/S0002-9947-1961-0130917-7 | MR 0130917 | Zbl 0109.26002

Affiliations: Mehdi Aaghabali (corresponding author), School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland; School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran, e-mail: maghabali@gmail.com; Mai Hoang Bien, Faculty of Mathematics and Computer Science, University of Science, VNU-HCM, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam, e-mail: mhbien@hcmus.edu.vn

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