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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...

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I like to think of this by building it up in stages. First take monomials in your subset $S$, so anything you can get by just multiplying elements of $S$ together. Then you can add these monomials up, so you get some finite sum of terms, each of which is a product of elements of $S$. Finally, you can take quotients of these, so for two such expressions $f$ and $g$, where $g\ne0$, you get the quotient $\frac{f}{g}=fg^{-1}$.

So the subfield generated by $S$ is exactly the set of these quotients $\frac{f}{g}$, where $f$ and $g$ are sums of monomials in $S$.

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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 $s+t, st \in K$.

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Yeah, I prefer this description too. I failed to point out that I already had this version handed to me. This way of thinking works in much greater generality for dealing with "subobjects." –  AsinglePANCAKE Sep 7 '12 at 12:52

It is the set of rational functions over elements of the generating set. I.e., every element has the form: $$\frac{p(s_1, \ldots, s_n)}{q(s_1, \ldots, s_n)}$$ where $p(X_1, \ldots X_n)$ and $q(X_1, \ldots X_n)$ are polynomials with integer coefficients in the indeterminates $X_1, \ldots, X_n$, for some $n$ and $s_1 \ldots s_n$ range over the generating set.

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What are the coefficients of the polynomials? I think all of them being $1$ except the constant term which is $0$ will be fine, or alternatively they could be in $S$, but this adds a lot of redundancy. Maybe other things work too. –  Matt Pressland Sep 7 '12 at 12:44
    
@MattPressland: thank you. The coefficients need to range over the integers. I have fixed my answer accordingly. –  Rob Arthan Nov 9 '13 at 22:47

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