especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts...

also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ring is a set endowed with two binary operations...

known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting...

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The...

ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with...

Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K[x1, ..., xn] over a field K. A Gröbner basis allows many important...

ideal of a multivariate polynomial ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered by the total...

coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called...

abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous...

mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by...

polynomials in X {\displaystyle X} form a ring denoted F [ X , X − 1 ] {\displaystyle \mathbb {F} [X,X^{-1}]} . They differ from ordinary polynomials...

commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z...

or zero of each polynomial in Jα More specifically, Jα is the kernel of the ring homomorphism from F[x] to E which sends polynomials g to their value...

Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization...

content by a unit of the ring of the coefficients (and the multiplication of the primitive part by the inverse of the unit). A polynomial is primitive if its...

group multiplication. the commutative algebra K[x] of all polynomials over K (see polynomial ring). algebras of functions, such as the R-algebra of all real-valued...

monic polynomials in a univariate polynomial ring over a commutative ring form a monoid under polynomial multiplication. Two monic polynomials are associated...

commutative ring. The rational, real and complex numbers form fields. If R {\displaystyle R} is a given commutative ring, then the set of all polynomials in the...

zeros of a set of polynomials in n variables. If A is such an algebraic set, one considers the commutative ring R of all polynomial functions A → K. The...

the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves...

In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z⟨X1...

mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of...

R_{i}=0} for i ≠ 0. This is called the trivial gradation on R. The polynomial ring R = k [ t 1 , … , t n ] {\displaystyle R=k[t_{1},\ldots ,t_{n}]} is...

integer-valued polynomials was described fully by George Pólya (1915). Inside the polynomial ring Q [ t ] {\displaystyle \mathbb {Q} [t]} of polynomials with rational...

for its applications, such as homological properties and polynomial identities. Commutative rings are much better understood than noncommutative ones. Algebraic...

field F is algebraically closed if every non-constant polynomial in F[x] (the univariate polynomial ring with coefficients in F) has a root in F. As an example...

Integer-valued polynomial Algebraic equation Factor theorem Polynomial remainder theorem See also Theory of equations below. Polynomial ring Greatest common...

a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules...

quotient ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } (which has n {\displaystyle n} elements). Now consider the ring of polynomials in the variable...