---- > [!definition] Definition. ([[rational function]]) > Let $R$ be a (commutative) [[ring]] (say, an [[integral domain]]). The [[field]] of **rational functions with coefficients in $R$** is the [[field of fractions]] of the [[polynomial 4|polynomial ring]] $R[x]$. This [[field]] is denoted $R(x)$. > Elements $R[x]$ are fractions of polynomials $\frac{p(x)}{q(x)}$ with $p(x),q(x) \in R[x]$ and $q(x) \neq 0$. The term *function* is inaccurate, since the 'function' $R\to R$ given by $a \mapsto \frac{p(a)}{q(a)}$ is not defined for all $a \in R$ (not for those for which $q(a)=0$, that is) and the function itself does not suffice to determine the element in $R(x)$ (Aluffi). ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```