[[Field theory MOC]]
# Category of fields

The **category of fields** $\cat{Fld}$ is the [[subcategory]] of $\Ring$ containing only [[Field|fields]]. #m/def/ring 
Since [[Field homomorphisms are injective|field homomorphisms are injective]],
the only morphisms are [[Field extension|field extensions]], and are thus [[Monomorphism|monic]].
By fixing a [[characteristic]] $p$, we may construct the subcategory 
[[Category of fields of characteristic p]], and have
 $$
\begin{align*}
\cat{Fld} = \coprod_{k = 0}^\infty \cat{Fld}_{p}
\end{align*}
$$


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#state/tidy| #lang/en | #SemBr