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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
Polynomial factoring algorithms use basic polynomial operations such as products, divisions, gcd, powers of one polynomial modulo another, etc. A multiplication of two polynomials of degree at most n can be done in O(n 2) operations in F q using "classical" arithmetic, or in O(nlog(n) log(log(n)) ) operations in F q using "fast" arithmetic.
[]: rings of polynomials in one variable with coefficients in a field. (The converse is also true, i.e. if A [ x ] {\displaystyle A[x]} is a PID then A {\displaystyle A} is a field.) Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form ( x k ) {\displaystyle (x^{k})} ,
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 ...
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.
If the localization S −1 A is a UFD, so is A. [1] (Note that the converse of this is trivial.) An integral domain A satisfies (ACCP) if and only if the polynomial ring A[t] does. [2] The analogous fact is false if A is not an integral domain. [3]
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).