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?