In Basic Category Theory by Tom Leinster, it is mentioned in Examples 2.1.3 that the forgetful functor $U:\mathbf{Field}\to\mathbf{Set}$ has no left adjoint. A proof is given later in Example 6.3.5, for which I don't have the knowledge yet.
My goal is now to intuitively understand where this fails. As a beginner, I feel like the left adjoint functor of $U$ should just be "the functor $F:\mathbf{Set}\to\mathbf{Field}$, where for a set $S$, we define $F(S)$ to be the "field generated by elements of $S$ in which no unexpected relations hold". For instance, if $x$ and $y$ are elements of $S$, then $F(S)$ will contain the elements $x$, $x+y$, $\frac xy$, some elements $0,1\not\in S$, and so on.
I assume that $F$ is not the left adjoint of $U$ because $F$ does not even exist. Is this correct? Why is this?
Best Answer
Your intuition is correct about what a left adjoint functor should be - freely generated by the set. Formally, this is a consequence of adjunction: each ring homomorphism $f:F(S)\to K$ from the free field on a set $S$ into a field $K$ is uniquely determined by the restriction map $f|_S:S\to K$.
Now if you want to try and see why there is no functor $F$ with this property, explore the consequences of this condition. What do you know about field homomorphisms?