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If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.
A polynomial ring over a UFD is a UFD. Indeed, by induction, it is enough to show [] is a UFD when is a UFD. Let [] be a non-zero polynomial. Now, [] ...
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 induction on the number of indeterminates. The main such properties are listed below.
The other class of Dedekind rings that is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k[C] of regular functions on C is a Dedekind domain.
The ring of polynomials in finitely-many variables over the integers or a field is Noetherian. Rings that are not Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings: The ring of polynomials in infinitely-many variables, X 1, X 2, X 3, etc.
Irreducible polynomials over finite fields are also useful for pseudorandom number generators using feedback shift registers and discrete logarithm over F 2 n. The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces , given by Moreau's necklace-counting function M q ( n ).
A symmetric algebra over a field (since every symmetric algebra is isomorphic to a polynomial ring in several variables over a field). Let k {\displaystyle k} be a field of characteristic not 2 and S = k [ x 1 , … , x n ] {\displaystyle S=k[x_{1},\dots ,x_{n}]} a polynomial ring over it.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.