Magnetic Field in Perfect Electric Conductor I know that one of Maxwell's equations states that the curl of the electric field is proportional to the time derivative of the magnetic field. We know that the electric field in a perfect electric conductor is 0, so why couldn't the magnetic field just be a constant instead of 0? 
Edit: There is no current everywhere in space.
 A: Maxwell's equations give us the divergence and the curl of the magnetic field in this case:
$$\nabla\cdot\mathbf{B}=0$$
$$\nabla\times\mathbf{B}=\mu_0\mathbf{J}$$
For a conductor with current flowing, there is obviously a nonzero, non-constant (in space) magnetic field with a finite curl. I gather that this is not the situation you asked about, since you asked specifically about differentiating between zero field and a constant field (i.e. constant in space). In this particular case, we have a conductor with no current flowing, which means:
$$\nabla\cdot\mathbf{B}=0$$
$$\nabla\times\mathbf{B}=0$$
There is a very important theorem in vector calculus (commonly called the Helmholtz theorem) that, in a particular formulation, states:

Any field that vanishes at infinity is uniquely determined by its divergence and curl.

In other words, if we find a field with zero divergence and curl everywhere that also vanishes at infinity, then we have found the only field with these properties.
Here's the important part: for any finite conductor, we expect the magnetic field to vanish at infinity. So we need a field that has zero divergence and zero curl everywhere which also vanishes at infinity. It turns out that $\mathbf{B}=0$ is exactly such a field, and by the theorem above, it is the only field configuration which can describe this situation.
The other important part: any infinite conductor is constructed as a limit of progressively larger finite conductors. Since the field is zero everywhere for all of the finite conductors, it's natural to choose the field to be zero everywhere for infinite conductors as well.
