Since there is no "free field generated by a set", it would seem that


1) there is no monad on Set whose algebras are exactly the fields


and


2) there is no Lawvere theory whose models in Set are exactly the fields


(Are 1) and 2) correct?)


Fields don't form a variety of algebras in the sense of universal algebra since the field axioms can´t be written as identities (since the axiom for multiplicative inverses has the restriction that the element be non-zero).


I guess fields are an algebraic theory in a more general universal algebra sense of being defined by operations on a single set with a set of first order axioms.


Is there any better sense in which they are algebraic or are fields just not really algebraic in nature?



In what sense are fields an algebraic theory?
1 and 2 are correct, for a simple reason. If *C* is a category satisfying either 1 or 2 then *C* has a terminal object. But there is no terminal object in the category of fields (and ring homomorphisms), because there are no maps between fields of different characteristic. 


For the same reason, the category of fields is not an essentially algebraic theory (mentioned in Andrew's answer). An **essentially algebraic theory** can be defined as, simply, a small category with finite limits. A **model** or **algebra** for an essentially algebraic theory *T* is a finite-limit-preserving functor *T* --> **Set**. (Of course, you can consider models in other finite limit categories too.) And the category of models always has a terminal object.


This embodies the idea that Andrew was describing, of a theory where some operations are only partially defined, but (and this is crucial!) the domain of definition is itself defined by equations. You can see some rough connection between finite limits and this intuitive idea if you consider pullbacks in **Set**. A pullback in **Set** is, after all, the set of pairs satisfying some equation.


I don't know in what sense the theory of fields *is* algebraic. It's partly because of its failure to be algebraic in any of the usual senses that one often chooses to work with commutative rings rather than fields, in algebraic geometry and in topos theory, for instance. 


