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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0310REJJY8]] of finite [[Krull dimension|MVGBS3J1P7]] $\\dim R$, and let $R[X]$ be the [[polynomial ring|0SDC934R52]] in one variable over $R$.\n\n1. In general, $\\dim R + 1 \\le \\dim R[X] \\le 2 \\dim R + 1$.\n2. If $R$ is a [[Noetherian ring|0670T59HGE]], then $\\dim R[X] = \\dim R + 1$.\n\n## Remarks\n\nFor the lower bound, a chain of [[prime ideals|00MHHZ9SNE]] $\\mathfrak p_0 \\subsetneq \\cdots \\subsetneq \\mathfrak p_n$ of $R$ gives the chain $\\mathfrak p_0[X] \\subsetneq \\cdots \\subsetneq \\mathfrak p_n[X] \\subsetneq \\mathfrak p_n[X] + (X)$ in $R[X]$.\nFor the upper bound, fix a prime $\\mathfrak p$ of $R$ and let $\\kappa(\\mathfrak p)$ be the fraction field of $R/\\mathfrak p$.\nThe primes of $R[X]$ that contract to $\\mathfrak p$ correspond to primes of $\\kappa(\\mathfrak p)[X]$, so at most two of them lie in one chain.\nIn the Noetherian case one shows that a prime $\\mathfrak P$ of $R[X]$ over $\\mathfrak p$ has height at most $\\operatorname{ht} \\mathfrak p + 1$.\nBy induction, for a field $k$ the ring $k[X_1, \\dots, X_n]$ has dimension $n$, and for a Noetherian ring $R$ of finite dimension, $\\dim R[X_1, \\dots, X_n] = \\dim R + n$.\nFor example, $\\dim \\Z[X] = 2$, with the maximal chain $0 \\subsetneq (2) \\subsetneq (2, X)$.\nWithout the Noetherian hypothesis the equality in clause 2 can fail: there are non-Noetherian rings with $\\dim R[X] \u003e \\dim R + 1$.\n\n## Explanations\n\n### L1 human\n\nThe Krull dimension counts how long a chain of prime ideals can be.\nAdding one polynomial variable adds exactly one to this count when the ring is Noetherian.\nThis matches geometry: a line has dimension $1$, the plane $2$, and $n$-space $n$.\n\n### L2 human\n\nThis result is the algebraic reason why affine $n$-space over a field has dimension $n$.\nTogether with [[Noether normalization|7Z28JY8QW4]] it shows that a finitely generated domain over a field has dimension equal to the transcendence degree of its fraction field.\nThe general bounds come from looking at the fibres of $\\operatorname{Spec} R[X] \\to \\operatorname{Spec} R$, which are spectra of polynomial rings in one variable over fields.\n",
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