Editing Polynomial ring (section)

== Univariate over a ring vs. multivariate ==
A polynomial in <math>K[X_1, \ldots, X_n]</math> can be considered as a univariate polynomial in the indeterminate <math>X_n</math> over the ring <math>K[X_1, \ldots, X_{n-1}],</math> by regrouping the terms that contain the same power of <math>X_n,</math> that is, by using the identity
:<math>\sum_{(\alpha_1, \ldots, \alpha_n)\in I} c_{\alpha_1, \ldots, \alpha_n} X_1^{\alpha_1} \cdots X_n^{\alpha_n}=\sum_i\left(\sum_{(\alpha_1, \ldots, \alpha_{n-1})\mid (\alpha_1, \ldots, \alpha_{n-1}, i)\in I} c_{\alpha_1, \ldots, \alpha_{n-1}} X_1^{\alpha_1} \cdots X_{n-1}^{\alpha_{n-1}}\right)X_n^i,</math>
which results from the distributivity and associativity of ring operations.

This means that one has an [[algebra isomorphism]]
:<math>K[X_1, \ldots, X_n]\cong (K[X_1, \ldots, X_{n-1}])[X_n]</math>
that maps each indeterminate to itself. (This isomorphism is often written as an equality, which is justified by the fact that polynomial rings are defined up to a ''unique'' isomorphism.)

In other words, a multivariate polynomial ring can be considered as a univariate polynomial over a smaller polynomial ring. This is commonly used for proving properties of multivariate polynomial rings, by [[mathematical induction|induction]] on the number of indeterminates.

The main such properties are listed below.

=== Properties that pass from {{math|''R''}} to {{math|''R''[''X'']}} ===
In this section, {{mvar|R}} is a commutative ring, {{mvar|K}} is a field, {{mvar|X}} denotes a single indeterminate, and, as usual, <math>\mathbb Z</math> is the ring of integers. Here is the list of the main ring properties that remain true when passing from {{mvar|R}} to {{math|''R''[''X'']}}.

* If {{mvar|R}} is an [[integral domain]] then the same holds for {{math|''R''[''X'']}} (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coefficients of the factors).
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are integral domains.
* If {{mvar|R}} is a [[unique factorization domain]] then the same holds for {{math|''R''[''X'']}}. This results from [[Gauss's lemma (polynomial)|Gauss's lemma]] and the unique factorization property of <math>L[X],</math> where {{mvar|L}} is the field of fractions of {{mvar|R}}.
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are unique factorization domains.
* If {{mvar|R}} is a [[Noetherian ring]], then the same holds for {{math|''R''[''X'']}}.
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are Noetherian rings; this is [[Hilbert's basis theorem]].
* If {{mvar|R}} is a Noetherian ring, then <math>\dim R[X] = 1+\dim R,</math> where "<math>\dim</math>" denotes the [[Krull dimension]].
**In particular, <math>\dim K[X_1,\ldots,X_n] = n</math> and <math>\dim \mathbb Z[X_1,\ldots,X_n] = n+1.</math>
* If {{mvar|R}} is a [[regular ring]], then the same holds for {{math|''R''[''X'']}}; in this case, one has <math display="block">\operatorname{gl}\, \dim R[X]= \dim R[X]= 1 + \operatorname{gl}\, \dim R=1+\dim R,</math> where "<math>\operatorname{gl}\, \dim</math>" denotes the [[global dimension]].
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math>  are regular rings, <math>\operatorname{gl}\, \dim \mathbb Z[X_1,\ldots,X_n] = n+1,</math> and <math>\operatorname{gl}\, \dim K[X_1,\ldots,X_n] = n.</math> The latter equality is [[Hilbert's syzygy theorem]].