A full subcategory $\mathbb{B}$ of a category $\mathbb{A}$ is reflective if the inclusion functor has a left adjoint $R$. The left adjoin is called the reflector of $\mathbb{B}$, and the unit of the adjuction $\eta$ satisfies the next universal property. For all $A\in\mathbb{A}$, if there is a morphism $f\colon A\longrightarrow B$ with $B\in \mathbb{B}$ then there is an unique morphism $\alpha\colon T(A)\longrightarrow B $ such that $\alpha\circ\eta_A=f$.
As in the category of the fields all the morphisms are monomorphisms. The morphism can be though as field extension. In particular we have that $Gal(T(F)/F)=\{1\}$ for any field $F$. This condition seems (for me) very restrictive. That's why I think there is no reflective subcategories besides the total.