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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.
This can be used for an equivalent definition of polynomial rings. The polynomial ring in X over K is equipped with an addition, a multiplication and a scalar multiplication that make it a commutative algebra. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if
The ring of Laurent polynomials over a field is Noetherian (but not Artinian). If R {\displaystyle R} is an integral domain , the units of the Laurent polynomial ring R [ X , X − 1 ] {\displaystyle R\left[X,X^{-1}\right]} have the form u X k {\displaystyle uX^{k}} , where u {\displaystyle u} is a unit of R {\displaystyle R} and k ...
This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be ...
2. A Hironaka decomposition is a representation of a ring as a finite free module over a polynomial ring or regular local ring. 3. Hironaka's criterion states that a ring that is a finite module over a regular local ring or polynomial algebra is Cohen–Macaulay if and only if it is a free module. Hodge 1. Named after W. V. D. Hodge 2.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
[]: 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 ...
Z[ω] (where ω is a primitive (non-real) cube root of unity), the ring of Eisenstein integers. Define f (a + bω) = a 2 − ab + b 2, the norm of the Eisenstein integer a + bω. K[X], the ring of polynomials over a field K. For each nonzero polynomial P, define f (P) to be the degree of P. [4] K[[X]], the ring of formal power series over the ...