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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]], let $S = k[x_1, \\ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], and fix a [[monomial order|E874NARWPV]] on $S$.\nFor a nonzero $f \\in S$, let $\\operatorname{LT}(f)$ be its leading term.\nLet $I$ be an [[ideal|096Z9MP2ME]] of $S$, and let $\\operatorname{LT}(I)$ be the ideal generated by the leading terms $\\operatorname{LT}(f)$ of all nonzero $f \\in I$.\nA finite subset $G = \\set{g_1, \\ldots, g_t}$ of $I \\setminus \\set{0}$ is a *Gröbner basis* of $I$ if\n$$(\\operatorname{LT}(g_1), \\ldots, \\operatorname{LT}(g_t)) = \\operatorname{LT}(I).$$\n\n## Remarks\n\n$\\operatorname{LT}(I)$ is a [[monomial ideal|4G25S7PJ22]].\nEquivalent forms: for every nonzero $f \\in I$, some $\\operatorname{LT}(g_i)$ divides $\\operatorname{LT}(f)$.\nOr: for every $f \\in S$, the remainder of $f$ under the [[multivariate division algorithm|T8EN3CAEQF]] by $G$ is zero if and only if $f \\in I$.\nA Gröbner basis of $I$ generates $I$, and every ideal has a Gröbner basis; this gives another proof of [[Hilbert's basis theorem|0YKA9J3XEE]].\nA Gröbner basis is *reduced* if each $\\operatorname{LT}(g_i)$ has coefficient $1$ and no monomial of $g_i$ is divisible by $\\operatorname{LT}(g_j)$ for $j \\ne i$.\nFor a given monomial order, each ideal has a unique reduced Gröbner basis.\n[[Buchberger's algorithm|4RH1D40WG1]] computes a Gröbner basis from any finite set of generators.\n\nExample: $G = \\set{x, y}$ is a Gröbner basis of $I = (x, y)$ in $k[x, y]$, for any monomial order.\nEvery nonzero $f \\in I$ has no constant term, so each of its monomials, in particular $\\operatorname{LT}(f)$, is divisible by $x$ or by $y$.\n\nNon-example: for the lexicographic order with $x \u003e y$, the set $G = \\set{x + y, \\; x}$ generates the same ideal $I$, and both leading terms equal $x$.\nBut $y = x - (x + y) \\in I$ and $\\operatorname{LT}(y) = y \\notin (x)$, so $G$ is not a Gröbner basis of $I$.\n",
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