# Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
division ring; rational identity; maximal subfield
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 $D.$
References:
[1] Aaghabali, M., Akbari, S., Bien, M. H.: 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] Albert, A. A., Muckenhoupt, B.: On matrices of trace zeros. Mich. Math. J. 4 (1957), 1-3. DOI 10.1307/mmj/1028990168 | MR 0083961 | Zbl 0077.24304
[3] Amitsur, S. A.: 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] Amitsur, S. A., Rowen, L. H.: Elements of reduced trace 0. Isr. J. Math. 87 (1994), 161-179. DOI 10.1007/BF02772992 | MR 1286824 | Zbl 0852.16012
[5] Beidar, K. I., Martindale, W. S., III, Mikhalev, A. V.: Rings with Generalized Identities. Pure and Applied Mathematics 196, Marcel Dekker, New York (1996). MR 1368853 | Zbl 0847.16001
[6] Chebotar, M. A., Fong, Y., Lee, P.-H.: 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] Chiba, K.: 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] Hai, B. X., Dung, T. H., Bien, M. H.: Almost subnormal subgroups in division rings with generalized algebraic rational identities. Available at https://arxiv.org/abs/1709.04774
[9] Lam, T. Y.: 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] Mahdavi-Hezavehi, M.: 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] Mahdavi-Hezavehi, M.: Commutators in division rings revisited. Bull. Iran. Math. Soc. 26 (2000), 7-88. MR 1828953 | Zbl 0983.16012
[12] Mahdavi-Hezavehi, M., Akbari-Feyzaabaadi, S., Mehraabaadi, M., Hajie-Abolhassan, H.: On derived groups of division rings. II. Commun. Algebra 23 (1995), 2881-2887. DOI 10.1080/00927879508825374 | MR 1332151 | Zbl 0866.16012
[13] Thompson, R. C.: 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

Partner of