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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 ]
In algebra, a multilinear polynomial [1] is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables.
An example of a principal ideal domain that is not a Euclidean domain is the ring [+], [6] [7] this was proved by Theodore Motzkin and was the first case known. [8] In this domain no q and r exist, with 0 ≤ | r | < 4 , so that ( 1 + − 19 ) = ( 4 ) q + r {\displaystyle (1+{\sqrt {-19}})=(4)q+r} , despite 1 + − 19 {\displaystyle 1+{\sqrt ...
Milos Uzan scored a season-high 19 points to help No. 6 Houston rally for a 62-58 victory over No. 13 Arizona in Big 12 play on Saturday afternoon at Tucson, Ariz. L.J. Cryer added 15 points for ...
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.