# Is there any non trivial reflective subcategory of the category of fields?

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.

• Your $TA$ is contained in every $B$ which contains $A$. In particular, it is contained in $A$. – Mariano Suárez-Álvarez Sep 17 '16 at 20:17

The category of perfect fields $k$, i.e. those of characteristic $0$ or of characteristic $p>0$ such that $k=k^p$, is an example. The left adjoint to the inclusion is given by adjoining all $p^r$-th roots.