---- > [!definition] Definition. ([[affine variety]]) > Let $k$ be an [[algebraically closed]] [[field]]. An [[algebraic set|(affine) algebraic set]] $X \subset k^{n}$, $X=V(S)$ for some $S \subset k[T_{1},\dots,T_{n}]$, is called an **affine variety** if it is [[irreducible algebraic set|irreducible]].[^1] Algebraically, [[affine variety|this means that]] the [[ideal]] $\langle S \rangle \subset k[T_{1},\dots,T_{n}]$ [[ideal generated by a subset|generated by]] $S$ is [[prime ideal|prime]], i.e., that the [[coordinate ring]] $k[T_{1},\dots,T_{n}] /I(X)$ is an [[integral domain]]. ^definition ---- #### [^1]: Some sources will treat 'algebraic set' and 'affine variety' as earnest synonyms, omitting the irreducibility requirement. ---- #### 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 > ``` Let $\mathbb{A}^{n}_{k}$ denote the $n$-dimensional [[affine space]] over an [[algebraically closed]] [[field]] $k$. An **affine variety** $V \subset \mathbb{A}^{n}_{k}$ is given by the vanishing of [[polynomial 4|polynomials]] $f_{1},\dots,f_{r} \in k[X_{1},\dots,X_{n}]$: $V=V(f_{1},\dots,f_{r})=\{ z \in \mathbb{A}^{n}_{k}:f_{1}(z)=\dots=f_{r}(z)=0 \}$ Equivalently, if $I=\langle f_{1},\dots,f_{r} \rangle \subset k[X_{1},\dots,X_{n}]$ is any [[ideal]], we set $\mathbb{V}(I):= \{ z \in \mathbb{A}^{n}_{k}: f(z)=0\ \fa f \in I \}.$