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  "body": "\n## Definition\n\nLet $k$ be a field and let $n \\ge 1$.\nThe monomials of the [[polynomial ring|0SDC934R52]] $k[x_1, \\ldots, x_n]$ are the products $x^a = x_1^{a_1} \\cdots x_n^{a_n}$ with $a \\in \\N^n$.\nA *monomial order* is a relation $\u003e$ on the set of monomials such that:\n\n1. $\u003e$ is a strict [[total order|0FRXBVVR03]].\n2. If $x^a \u003e x^b$, then $x^a x^c \u003e x^b x^c$ for every $c \\in \\N^n$.\n3. $\u003e$ is a [[well-order|01CGSJ0GNQ]]: every nonempty set of monomials has a smallest element.\n\n## Remarks\n\nGiven clauses 1 and 2, clause 3 is equivalent to $x^a \u003e 1$ for every $a \\ne 0$; this uses Dickson's lemma.\nEquivalently, a monomial order is a total order on $\\N^n$ that is compatible with addition and has $0$ as its least element.\nFor a nonzero polynomial $f$, the leading monomial $\\operatorname{LM}(f)$ is the largest monomial that occurs in $f$, and the leading term $\\operatorname{LT}(f)$ is that monomial with its coefficient.\n\nExample: the [[lexicographic order|0B68KNPGCG]]: $x^a \u003e x^b$ if the first nonzero entry of $a - b$ is positive.\nIt is total, it is compatible since $(a + c) - (b + c) = a - b$, and it is a well-order on $\\N^n$.\n\nNon-example: for $n = 1$, order the monomials by $x^a \u003e x^b$ if $a \u003c b$.\nThis is a total order compatible with multiplication, but $x \u003e x^2 \u003e x^3 \u003e \\cdots$ is an infinite descending chain, so clause 3 fails.\n",
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