  We show that, within a linear approximation of BCS theory, a weak homogeneous
magnetic field lowers the critical temperature by an explicit constant times
the field strength, up to higher order terms. This provides a rigorous
derivation and generalization of results obtained in the physics literature
from WHH theory of the upper critical magnetic field. A new ingredient in our
proof is a rigorous phase approximation to control the effects of the magnetic
field.
