On the complexity of the isomorphism relation for fields of finite transcendence degree

Confirming a conjecture of Hjorth and Kechris (Ann. Pure Appl. Logic 82 (1996) 221–272), we prove that the isomorphism relation for fields of finite transcendence degree is a universal essentially countable Borel equivalence relation. We also prove that the theory of fields of finite transcendence degree does not admit canonical models.