Example 9.3.5 (Field of rational functions). If $k$ is a field, then we can consider the field $k(x)$ of rational functions over $k$. This is the quotient field of the polynomial ring $k[x]$. In other words, it is the set of quotients $F/G$ for $F, G \in k[x]$, $G \not= 0$ with the obvious equivalence relation.
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