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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
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.
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig.
[]: the ring of all polynomials with integer coefficients. It is not principal because 2 , x {\displaystyle \langle 2,x\rangle } is an ideal that cannot be generated by a single polynomial. K [ x , y , … ] , {\displaystyle K[x,y,\ldots ],} the ring of polynomials in at least two variables over a ring K is not principal, since the ideal x , y ...
In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID. Algebraic structures; Group-like. ... the polynomial (which is ...
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
A is a UFD. A satisfies (ACCP) and every irreducible of A is prime. A is a GCD domain satisfying (ACCP). The so-called Nagata criterion holds for an integral domain A satisfying (ACCP): Let S be a multiplicatively closed subset of A generated by prime elements. If the localization S −1 A is a UFD, so is A. [1] (Note that the converse of this ...