Alphabeta Math
CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-09-01
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A polynomial ring in n variables over a field has dimension n

Statement

Let k be a field and let n≥0. Then

dim⁡k[x1,…,xn]=n.

Facts & Assumptions

Given: A field k and an integer n≥0.

[L1]

Adjoining one polynomial variable to a finite-dimensional Noetherian ring raises dimension by one (A Noetherian polynomial ring has dimension one larger).

Proof

technique · induction on the number of variables
1.1

For n=0, the ring is the field k, whose only prime ideal is (0), so its dimension is 0.

basegiven
1.2

If dim⁡k[x1,…,xn]=n, then k[x1,…,xn+1]≅k[x1,…,xn][xn+1], so [L1] gives dimension n+1.

L1ih
2.1

Therefore dim⁡k[x1,…,xn]=n for every n≥0.

step 1.1step 1.2discharge-induction∎

Depends on

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources