TITLE: What is a field?
QUESTION [16 upvotes]: I've always wondered about what a field is meant to represent. For example, group automorphisns naturally represent symmetry in many areas.
I'm not looking for a solid answer, just an idea.

REPLY [10 votes]: From a semi-category-theoretic point of view, fields are to commutative rings as simple groups are to groups.
Specifically, a group, $G$, is simple if any only if any homomorphism $G\rightarrow H$ is either a monomorphism or trivial.
In the category of commutative rings, $F$ is a field if and only if any ring homomorphism $F\rightarrow R$ is either a monomorphism or trivial.