TITLE: Given any computable number, is there any algorithm to decide whether it is transcendental?
QUESTION [21 upvotes]: Given any computable number $a_c$, is there any algorithm to decide whether it is transcendental?
Definition of “computable number”: According to Ming Li and Vitanyi, a real number $x=0.x_1x_2\ldots$ is lower semicomputable if the set of rationals below $x$ is recursively enumerable. A number $−x$ is upper semicomputable if $x$ is lower semicomputable. A number $x$ is computable (equivalently, recursive) if it is both lower semicomputable and upper semicomputable. I think that this is equivalent to the Turing definition.

REPLY [4 votes]: To add a small layer of emphasis to what's been written so far: you say 'given a computable number $a_c$', but — unlike the case with using integers as input — you can't really be given a number per se; instead, what you're given as input has to be seen as (equivalent to) a program for computing the number.  Viewed in this light, it's no surprise that Rice's theorem comes into play.
By contrast, in a computation model where reals (and integers) are 'atomic' and the atomic operations are e.g. addition/subtraction, multiplication/division, and comparison, then the set of algebraic numbers is c.e.; one can simply dovetail evaluation of all of the possible integer-valued polynomials to test whether the number in question is a root.  (I believe that in this model transcendentality is non-computable, and in fact that it's the equivalent of $\Pi_1^0$-complete, but don't hold me to that.)