English Dictionary

polynomial basis

名詞

有限体

(algebra, ring theory) A basis of a polynomial ring (said ring being viewed either as a vector space over the field of coefficients or as a free module over the ring of coefficients).

日本語の意味

（代数学、環論）多項式環の基底。すなわち、係数体上のベクトル空間または係数環上の自由加群として見なす多項式環における基底を指す。

When studying the structure of k[x], the professor emphasized that choosing a polynomial basis simplifies many proofs about linear independence.

k[x] の構造を研究する際に、教授は多項式環の基底を選ぶことが線形独立性に関する多くの証明を簡素化すると強調した。

意味（1）

(algebra, ring theory) A basis of a polynomial ring (said ring being viewed either as a vector space over the field of coefficients or as a free module over the ring of coefficients).

If T(s)=M(s)D(s)⁻¹ is a RCMFD of T(s), then M(s) is a polynomial basis for 𝔛_t. If Q(s) is a greatest right divisor of M(s) then T(s)=◌̅M(s)Q(s)D(s)⁻¹, where ◌̅M(s) is a least degree polynomial basis of 𝔛_t [15].

意味（2）

(algebra, field theory, cryptography, of a finite field) Specifically, a basis, of the form {1, α, ..., αⁿ⁻¹}, of a finite extension F_(qⁿ) of a Galois field F_q, where α is a primitive element of F_(qⁿ) (i.e., a root of a degree-n primitive polynomial over F_q).

When using polynomial bases, the first stage in computing the product of two elements of 𝔽2ⁿ is the multiplication of two polynomials of degree at most n-1 in 𝔽𝟚[x].

polynomial bases （ plural ）