Multivariate polynomial ring over a field

Definition

The multivariate polynomial ring over a field is defined as a polynomial ring in (finitely) many variables over a field. If we denote the underlying field by $k$, and the variables by $x_1,x_2,\dots ,x_n$, then the polynomial ring is denoted $k[x_1,x_2,\dots ,x_n]$.

Properties

The multivariate polynomial ring over a field is a unique factorization domain as well as a Noetherian domain. When there is more than one variable, it is not a Euclidean domain or a principal ideal domain.

Extra structure

Further information: multivariate polynomial ring#extra structure

The multivariate polynomial ring $k[x_1,x_2,\dots ,x_n]$ can be viewed as a $k$-algebra. It also has the following additional structures:

• Graded ring structure: In particular, it is a graded $k$-algebra where the $d^{th}$ component is the vector space generated by monomials of degree $d$
• Filtered ring structure: In particular, it is a filtered $k$-algebra where the $d^{th}$ filtration is the subspace of polynomials of degree at most $d$

Ideals and quotients

Quotients of the polynomial ring in $n$ variables, are precisely the same as algebras generated by $n$ elements as $k$-algebras. The structure of these is in general fairly complicated.

Spectrum

Further information: Spectrum of multivariate polynomial ring over a field, spectrum of multivariate polynomial ring over an algebraically closed field

Max-spectrum

Further information: Max-spectrum of multivariate polynomial ring over a field, max-spectrum of multivariate polynomial ring over an algebraically closed field