Abstract. The famous theorem of Amitsur characterizes finite-dimensional central division algebras over a given field that have the same splitting fields, including infinite-dimensional ones. The situation changes dramatically if one allows only finite-dimensional splitting fields or just the maximal subfields of the division algebras at hand. To quantify these issues, one defines the genus gen(D) of a central division algebra D of degree n over a field K as the set of classes [D'] in the Brauer group Br(K) represented by a central division K-algebra D' of degree n having the same maximal subfields as D. I will review the results on gen(D) obtained in the last several years, including the finiteness theorem for gen(D) when K is a finitely generated field. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic K-groups and report on some recent progress in this direction.
On the notion of genus for division algebras and algebraic groups
Amitsur Memorial Symposium 2020, Open University of Israel