Polynomial ring

Japanese: 多項式環 - たこうしきかん

Generalizing the idea of a polynomial in x with real coefficients, we can write a ₀ x ⁿ + a ₁ x ^{n -1} +……+ a _n for a commutative field k and a letter x.
An expression of the form is called a polynomial in x over k . Let us denote the set of polynomials in x over k by k [ x ], and call this k [ x ] a polynomial ring over k . Sum and product are defined in the usual way for polynomials, and k [ x ] becomes a commutative ring.

An important aspect of the theory of polynomial rings is the uniqueness theorem for decomposition into irreducible polynomials. If a polynomial f ( x ) over a commutative field k can be expressed as (1) f (x) = g ( x ) h ( x ) in terms of polynomials g ( x ) and h ( x ) also over k , and (2) g ( x ) and h ( x ) have lower degrees than f ( x ), then f ( x ) is said to be reducible over k . If (1) and (2) are not possible, then f(x) is said to be irreducible over k .

Irreducible polynomials in a polynomial ring are like prime numbers in a ring of integers. Just as every integer can be uniquely expressed as a product of powers of prime numbers (e.g., 100=2 ^{^2} × 5 ^{^2} ), it has been proven that every polynomial can be uniquely expressed as a power of an irreducible polynomial, which is the basis of the theory of algebraic equations.

Furthermore, a rational expression over k is an expression that can be expressed as f ( x )/ g ( x ) for polynomials f ( x ), g (x) over k (where g ( x ) ≠ 0). The set of rational expressions over a commutative field k is also a commutative field, which is called the field of rational functions.

[Terada Fumiyuki]

[Reference item] | Algebraic equations

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

実数を係数とするxの多項式の考えを一般化して、可換体kと文字xに対して
　　a₀xⁿ+a₁x^n-1+……+a_n
という形の式をk上xの多項式という。k上xの多項式全体をk［x］で表し、このk［x］をk上の多項式環という。多項式に対しては通常の方法で和と積が定義され、k［x］は可換環になる。

　多項式環の理論のなかで重要なのは、既約な多項式への分解への一意性の定理である。可換体k上の多項式f(x)が、同じくk上の多項式g(x),h(x)によって
（1）f(x)=g(x)h(x)と表される
（2）g(x),h(x)はf(x)より低次数
であるとき、f(x)はk上可約であるといわれ、（1）、（2）のようにはできないとき、k上既約であるといわれる。