Genus of division algebras over fields with infinite transcendence degree

We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let K be a finite field extension of a field which is a purely transcendental extension of infinite transcendence degree of some subfield. We show that if \({\mathcal D}\) is a central division K-algebra, then \(\textbf{gen}({\mathcal D})\) consists of Brauer classes \([{\mathcal D}']\) such that \([{\mathcal D}]\) and \([{\mathcal D}']\) generate the same subgroup of \(\text {Br} (K)\). In particular, the genus of any division K-algebra of exponent 2 is trivial. Note that the family of such fields is closed under finitely generated extensions. Moreover, if \(\text {char}(K) \ne 2\), we prove that the genus of a simple algebraic group of type \(\textrm{G}_2\) over such a field K is trivial.