Since there is no "free field generated by a set", it would seem that
- there is no monad on Set whose algebras are exactly the fields
and
- 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?
