TITLE: Smallest Subfield Generated by a set
QUESTION [5 upvotes]: So I'm essentially trying to find an explicit description of the smallest subfield generated by a subset of the field.  I know that if s is an element of the subset, we must also have its additive and multiplicative inverse.  Is there a succint way of describing the subfield set-theoretically?  Again, the intuition seems clear, but I'm struggling with a formal description...

REPLY [9 votes]: There are two ways to consider a subfield $K$ of a field $F$ generated by a subset $S \subset F$.  The "building up" approach has already been described, so here is the "top-down" approach (I find this more useful in practice).  
We can consider all subfields $L \subset F$ such that $S \subset L \subset F$.  Then $K = \bigcap_{L \in \mathcal{L}} L$, where $\mathcal{L} = \{ L | L \subset F \text{ is a subfield with } S \subset L \}$.  That is, $K$ is the intersection of all the subfields of $F$ which contain $S$.
To see that $K$ is truly a field, we need to notice that it contains all the right things.  Certainly $0,1 \in K$ and if $s \in K$, then $s^{-1}$ and $-s$ are in any field containing $s$, so they are in $K$.  Similarly, if $s, t \in K$ then $s+t, st \in K$.