= Polynomial-ring height and dimension formula
{title2=$\operatorname{ht}P+\dim(k[x_1,\ldots,x_n]/P)=n$}
For a <prime ideal> $P$ of a <polynomial ring> over a <field>, the quotient's <Krull dimension> is the transcendence degree of its <fraction field>, and its height is $n$ minus that degree. This dimension theorem identifies algebraic codimension with the length of prime chains. It must not be applied indiscriminately to every Noetherian <ring> without its polynomial-ring hypotheses.
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