  I prove several independence results in the choiceless ZF+DC theory which
separate algebraic and non-algebraic consequences of the axiom of choice. As an
example, let E be an equivalence relation resulting from a turbulent Polish
group action, let X be a Polish field, and let F be a countable subfield. It is
consistent with the choiceless theory ZF+DC that X has a transcendence basis
over F and E does not have a selector.
