V2EX irreducible polynomial

Irreducible Polynomial

释义 Definition

不可约多项式:在给定的系数范围(例如某个整数)内,不能分解成两个次数都大于 0 的多项式之积的多项式。常见语境是“在某个域上不可约”。(在不同系数范围下,同一个多项式可能可约或不可约。)

发音 Pronunciation (IPA)

/rdjusbl plnomil/

例句 Examples

An irreducible polynomial cannot be factored into lower-degree polynomials over the same field.
不可约多项式在同一个域上不能分解为更低次数多项式的乘积。

Over the rational numbers, \(x^2+1\) is an irreducible polynomial, but over the complex numbers it factors as \((x-i)(x+i)\).
在有理数范围内,\(x^2+1\) 是不可约多项式;但在复数范围内,它可以分解为 \((x-i)(x+i)\)。

词源 Etymology

irreducible 来自拉丁语词根 reducere(“带回、化简”)及否定前缀 *ir-*,整体含义是“不能再化简/不能再分解”。polynomial 由 *poly-*(“多”)与 -nomial(与“项/名称”相关的构词成分)组合而来,表示“由多构成的表达式”。合在一起即“不能再分解的多项式”。

相关词 Related Words

文学与著作 Literary Works

  • Algebra Serge Lang(代数学教材中讨论不可约多项式与域扩张时频繁出现)
  • Abstract Algebra David S. Dummit & Richard M. Foote(在多项式环、可约性判别、分裂域章节出现)
  • Algebra Michael Artin(在域、环与多项式分解相关章节出现)
  • Ideals, Varieties, and Algorithms Cox, Little & O’Shea(在 Grbner 基与多项式理想的语境中涉及不可约性)
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