---- Let $k \subset \Omega$ be [[field|fields]] with $\Omega$ [[algebraically closed]]. > [!definition] Definition. ([[irreducible algebraic set]]) > An [[algebraic set]] $X \subset \Omega^{n}$ is **irreducible** if $X \neq \emptyset$ and $X$ is not equal to the union of two proper algebraic subsets of $X$. ^definition > [!equivalence] > An [[algebraic set]] $X$ is irreducible if and only if the [[ideal]] $I(X) \subset k[T_{1},\dots,T_{n}]$ is [[prime ideal|prime]], that is, if and only if the [[coordinate ring]] $k[T_{1},\dots,T_{n}] / I(X)$ is an [[integral domain]]. This features (and is proved as part of) [[Hilbert's geometry-algebra correspondence]]. ^equivalence > [!basicexample] > The algebraic subset of $\mathbb{R}^{2}$ defined by the vanishing of $(X-Y)(Y-X)$ is not irreducible. This can be seen by examining the coordinate ring $k[X,Y] / \langle (X-Y)(Y-X) \rangle$. For example, > - $f(X,Y):=Y+X$ has $f(1,1)=2\neq 0$, and > - $g(X,Y):=Y-X$ has $g(1,1)=-2 \neq 0$. > But $fg=0$. So the coordinate ring must not be an [[integral domain]]. > > ![[CleanShot 2025-05-03 at [email protected]]] > > ---- #### ---- #### 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 > ```