Geodesic
Division algebras with the same maximal subfields
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 70 (2015) no. 1, pp. 83-112

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This is a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. Also discussed are various generalizations of this problem and some of its applications. In the last section the problem is extended to the context of absolutely almost simple algebraic groups. Bibliography: 51 titles.
Keywords: division algebra, unramified Brauer group, semisimple algebraic groups. 
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V. I. Chernousov; A. S. Rapinchuk; I. A. Rapinchuk. Division algebras with the same maximal subfields. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 70 (2015) no. 1, pp. 83-112. http://geodesic.mathdoc.fr/item/RM_2015_70_1_a2/