polynomial ring wikipedia - EAS
- In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.en.wikipedia.org/wiki/Polynomial_ring
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Xem tất cả trên WikipediaPolynomial ring - Wikipedia
https://en.wikipedia.org/wiki/Polynomial_ringIn mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often,
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Xem thêmThe polynomial ring, K[X], in X over a field (or, more generally, a commutative ring) K can be defined in several equivalent ways. One of them is to define K[X] as the set of expressions, called polynomials in X, of the form
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Xem thêmIf K is a field, the polynomial ring K[X] has many properties that are similar to those of the ring of integers Most of these similarities result from the similarity between the long division of integers and the long division of polynomials.
Most of the properties of...
Xem thêmGiven n symbols called indeterminates, a monomial (also called power product)
is a formal product of these indeterminates, possibly...
Xem thêmA polynomial in can be considered as a univariate polynomial in the indeterminate over the ring by regrouping the terms that contain the same power of that is, by using the identity
which results from the distributivity and associativity of ring...
Xem thêmPolynomial rings in several variables over a field are fundamental in invariant theory and algebraic geometry. Some of their properties, such as those described above can be reduced to the
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Xem thêmPolynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, noncommutative polynomial rings, skew polynomial rings, and polynomial rigs.
Infinitely many variables...
Xem thêmVăn bản Wikipedia theo giấy phép CC-BY-SAMục này có hữu ích không?Cảm ơn! Cung cấp thêm phản hồiRing of polynomial functions - Wikipedia
https://en.wikipedia.org/wiki/Ring_of_polynomial_functionsIn 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 k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V.
The explicit definition of the ring can be given as follows. If is a polynomial ring, then we can view as coordinate functions on ; i.e., when This suggests the following: given a vector space V, let k[V] b…Wikipedia · Nội dung trong CC-BY-SA giấy phépPolynomial identity ring - Wikipedia
https://en.wikipedia.org/wiki/Polynomial_identity_ringXem thêm trên en.wikipedia.org- Any subring or homomorphic imageof a PI-ring is a PI-ring.
- A finite direct productof PI-rings is a PI-ring.
- A direct product of PI-rings, satisfying the same identity, is a PI-ring.
- It can always be assumed that the identity that the PI-ring satisfies is multilinear.
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Polynomial - Wikipedia
https://en.wikipedia.org/wiki/PolynomialXem thêm trên en.wikipedia.org
The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomialwas first used in the 17th century.Twisted polynomial ring - Wikipedia
https://en.wikipedia.org/wiki/Twisted_polynomial_ring- Let k {\displaystyle k} be a field of characteristic p {\displaystyle p} . The twisted polynomial ring k { τ } {\displaystyle k\{\tau \}} is defined as the set of polynomials in the variable τ {\displaystyle \tau } and coefficients in k {\displaystyle k} . It is endowed with a ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation τ …
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polynomial ring - Wiktionary
https://en.wiktionary.org/wiki/polynomial_ringNoun [] polynomial ring (plural polynomial rings) A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K.1998, Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, page 270,
Ring (mathematics) - Wikipedia
https://en.wikipedia.org/wiki/Ring_(mathematics)Dedekind The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties.
Polynomring – Wikipedia
https://de.wikipedia.org/wiki/PolynomringWenn ein kommutativer Ring mit einer ist, dann ist der Polynomring [] die Menge aller Polynome mit Koeffizienten aus dem Ring und der Variablen zusammen mit der üblichen Addition und Multiplikation von Polynomen. Davon zu unterscheiden sind in der abstrakten Algebra die Polynomfunktionen, nicht zuletzt, weil unterschiedliche Polynome dieselbe Polynomfunktion …
Irreducible polynomial - Wikipedia
https://en.wikipedia.org/wiki/Irreducible_polynomialA polynomial with integer coefficients, or, more generally, with coefficients in a unique factorization domain R, is sometimes said to be irreducible (or irreducible over R) if it is an irreducible element of the polynomial ring, that is, it is not invertible, not zero, andR.
Primitive part and content - Wikipedia
https://en.wikipedia.org/wiki/Primitive_part_and_contentFor factoring a multivariate polynomial over a field or over the integers, one may consider it as a univariate polynomial with coefficients in a polynomial ring with one less indeterminate. Then the factorization is reduced to factorizing separately the primitive part and the content.
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