---- > [!definition] Definition. ([[field of rational functions of an affine variety]]) > Let $X$ be an [[affine variety]]. Recalling that our [[affine variety|affine varieties]] are [[irreducible algebraic set|irreducible]] [[algebraic set|algebraic sets]], [[Hilbert's geometry-algebra correspondence|we know]] the [[coordinate ring]] $k[X]$ of functions on $X$ is an [[integral domain]]. Its [[field of fractions]] is called the **field of rational functions on $X$** or **function field of $X$**, denoted $k(X):=\text{Frac }k[X]$. > > We call a rational function $h \in k(X)$ **regular at $p \in X$** if it can be written as a quotient $\frac{f}{g}$ with $f,g \in k[X]$, $g(p) \neq 0$. Can show that $\{ f \in k(X): f \text{ regular everywhere} \}=k[X],$ > reconciling the two notions of regularity. This means on can reconstruct an affine variety (up to isomorphism) from its ring of everywhere regular rational functions. ^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 > ```