“polynomial ring”的概念、定义、翻译、参考文献-科学参考

polynomial ring

Mathematics

ℝ[x] denotes the ring of polynomials with real coefficients in a variable x. The ring is commutative and has a multiplicative identity, the constant polynomial 1. Addition is performed coefficient-wise and products are formed by

$(\sum_{i = 0}^{m} a_{i} x^{i}) (\sum_{j = 0}^{n} b_{i} x^{i}) = (\sum_{k = 0}^{m + n} c_{k} x^{k})$

where $c_{k} = \sum_{i = 0}^{k} a_{i} b_{k - i}$ .

Given a commutative ring R with a multiplicative identity 1, a polynomial ring R[x] may be similarly defined. Factorization may be non-standard, though; note

$x^{2} - 1 = (x - 1) (x + 1) = (x - 3) (x - 5) in ℤ_{8} [x]$

and that x²−1 = 0 has four roots in ℤ₈. If R is an integral domain, then the formula

$deg (f g) = deg (f) + deg (g)$

holds. If R is a UFD, then R[x] is a UFD. If R is a field, then R[x] is a Euclidean domain. Polynomials rings such as R[x₁, x₂,…, x_n] in several (or infinitely many) variables may similarly be defined.

单词	polynomial ring
释义	polynomial ring Mathematics ℝ[x] denotes the ring of polynomials with real coefficients in a variable x. The ring is commutative and has a multiplicative identity, the constant polynomial 1. Addition is performed coefficient-wise and products are formed by $(\sum_{i = 0}^{m} a_{i} x^{i}) (\sum_{j = 0}^{n} b_{i} x^{i}) = (\sum_{k = 0}^{m + n} c_{k} x^{k})$ where $c_{k} = \sum_{i = 0}^{k} a_{i} b_{k - i}$ . Given a commutative ring R with a multiplicative identity 1, a polynomial ring R[x] may be similarly defined. Factorization may be non-standard, though; note $x^{2} - 1 = (x - 1) (x + 1) = (x - 3) (x - 5) in ℤ_{8} [x]$ and that x²−1 = 0 has four roots in ℤ₈. If R is an integral domain, then the formula $deg (f g) = deg (f) + deg (g)$ holds. If R is a UFD, then R[x] is a UFD. If R is a field, then R[x] is a Euclidean domain. Polynomials rings such as R[x₁, x₂,…, x_n] in several (or infinitely many) variables may similarly be defined.
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