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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 P with coefficients in a UFD R is then said to be primitive if the only elements of R that divide all coefficients of P at once are the invertible elements of R; i.e., the gcd of the coefficients is one. Primitivity statement: If R is a UFD, then the set of primitive polynomials in R[X] is closed under
An example of a principal ideal domain that is not a Euclidean domain is the ring [+ ... [X, Y] of polynomials in 2 variables is a UFD but is not a PID.
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.
A constant polynomial is either the zero polynomial, or a polynomial of degree zero. A nonzero polynomial is monic if its leading coefficient is 1. {\displaystyle 1.} Given two polynomials p and q , if the degree of the zero polynomial is defined to be − ∞ , {\displaystyle -\infty ,} one has
A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain. A valuation ring is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian.
The International Boxing Association said Monday it will file criminal complaints against the International Olympic Committee in the U.S., France and Switzerland. The Swiss-based IOC allowing ...
For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...