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On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k [ x , y , z ]/( x 2 + y 3 + z 7 ) at the prime ideal ( x , y , z ) then R is a local ring that is a UFD, but the formal power series ring R [[ X ]] over R is not 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
Over the complex numbers, the irreducible factors (those that cannot be factorized further) are all of degree one, while, over the real numbers, there are irreducible polynomials of degree 2, and, over the rational numbers, there are irreducible polynomials of any degree. For example, the polynomial is irreducible over the rational numbers, is ...
The real root of the polynomial for −23 is the reciprocal of the plastic ratio (negated), while that for −31 is the reciprocal of the supergolden ratio. The polynomials defining the complex cubic fields that have class number one and discriminant greater than −500 are: [ 5 ]
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 ...
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.
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 ...
I corrected the example [,] / (+), which was described as a non-UFD. It is a UFD. With this correction, however, the exposition is awkward. It claims that most factor rings are not UFD's and then gives an example of one which is a UFD. Perhaps someone should write up a nice (and correct) counterexample.