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Basically, I'm trying to show that the field of formal Laurent series is the field of fractions for the ring of power series. My question here is, how exactly does one prove that a given field is a field of fractions? Not necessarily for this case, but what does one have to verify for in general?
I apologize for the somewhat elementary question, but I've looked elsewhere but can't find anything that helps. We didn't really go over this in class.
For a field $K$ to be a field of fractions of an integral domain $R$, it is necessary and sufficient that (i) $R \subset K$ and (ii) For all $k \in K$, there exists $s \in R$ with $s \neq 0$ such that $k \cdot s \in R$ (i.e. if we write $r = k \cdot s$ then $k = \frac{r}{s}$).
Let us verify the claim. As you mention in the comments, the definition of a field of fractions is being the smallest field $K$ such that $R \subset K$. So let's assume that $R \subset K$ and show: $$K \text{ is the field of fractions of } R \Leftrightarrow \text{Condition (ii) given above}.$$
($\Rightarrow$) Suppose that $K$ is the field of fractions of $R$. Define $$K'= \left\{k \in K : \text{ there exists } r \in R \text{ such that } r \neq 0 \text{ and } k \cdot r \in R\right\}.$$ Then one can check (you should do so!) that $K'$ is a field. Certainly $K'$ contains $R$. Therefore since $K$ is the field of fractions (the smallest field that contains $R$) and $K' \subset K$, we get that $K' = K$. Therefore $K$ satisfies condition (ii).
($\Leftarrow$) Now let $K$ be a field with $R \subset K$ and $K$ satisfying condition (ii). We will show $K$ is the field of fractions of $R$. Denote the field of fractions of $R$ by $F(R)$. To prove the claim, we will show that for any $k \in K$, $k \in F(R)$. Indeed, then since $F(R)$ is the smallest field containing $R$, and $K \subset F(R)$, it must be that $K = F(R)$.
So let $k \in K$, and by condition (ii) there exists $s \in R$ with $s \neq 0$ such that $k \cdot s \in R$. So write $k \cdot s = r$, and hence (in $K$) we have $k = rs^{-1}$. But $r \in F(R)$ and $s^{-1} \in F(R)$ so therefore $k = rs^{-1} \in F(R)$. This proves that $K \subset F(R)$, as claimed.
If you use the Wikipedia definition. (field of fractions is the smallest field containing a given integral domain), you need to show that Laurent series form a field (easy), that they contain power series (easy), and finally they are the smallest. Since any such field has to contain $1/z$ and $z$ the rest follow by linearity.
