  We show that the embeddability relations for countable quandles and for
countable fields of any given characteristic other than 2 are maximally complex
in a strong sense: they are invariantly universal. This notion from the theory
of Borel reducibility states that any analytic quasi-order on a standard Borel
space essentially appears as the restriction of the embeddability relation to
an isomorphism-invariant Borel set. As an intermediate step we show that the
embeddability relation of countable quandles is a complete analytic
quasi-order.
