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  "body": "\n## Definition\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $n \\ge 0$.\nGive $k^n$ the Zariski topology, whose closed sets are the [[algebraic sets|VHFMQQYSTN]].\nAn *affine variety* in $k^n$ is an algebraic set $X \\subseteq k^n$ that is [[irreducible|SY2MWC434Z]].\nThis means that $X$ is nonempty and that $X$ is not the union of two algebraic sets strictly smaller than $X$.\n\n## Remarks\n\nAn algebraic set $X$ is irreducible if and only if its [[ideal|HH6VEVM6Z0]] $I(X)$ is a [[prime ideal|00MHHZ9SNE]].\nFor instance, if $X$ is irreducible and $fg \\in I(X)$, then $X = (X \\cap V(f)) \\cup (X \\cap V(g))$, and irreducibility forces $f \\in I(X)$ or $g \\in I(X)$.\nWith [[Hilbert's Nullstellensatz|D85VZWS9QK]], affine varieties in $k^n$ correspond to prime ideals of $k[x_1, \\dots, x_n]$.\n\nExample: the parabola $X = V(y - x^2) \\subseteq k^2$ is an affine variety.\nIts ideal is $(y - x^2)$, and $k[x, y]/(y - x^2) \\cong k[x]$ is an integral domain, so the ideal is prime.\n\nNon-example: $V(xy) \\subseteq k^2$ is an algebraic set but not an affine variety.\nIt is the union of the two axes, each strictly smaller than $V(xy)$.\n\nMany texts, especially those based on schemes, call any algebraic set an affine variety, or use the term for a variety isomorphic to a closed subvariety of some $k^n$.\n",
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