  We show that for both the unary relation of transcendence and the finitary
relation of algebraic independence on a field, the degree spectra of these
relations may consist of any single computably enumerable Turing degree, or of
those c.e. degrees above an arbitrary fixed $\Delta^0_2$ degree. In other
cases, these spectra may be characterized by the ability to enumerate an
arbitrary $\Sigma^0_2$ set. This is the first proof that a computable field can
fail to have a computable copy with a computable transcendence basis.
