irreducible

== English == === Etymology === From ir- +‎ reducible. === Pronunciation === IPA(key): /ˌɪɹɪˈdjuːsɪbəl/ === Adjective === irreducible (not comparable) Not able to be reduced or lessened. Antonym: unincreasable Not able to be brought to a simpler or reduced form. (mathematics, of a polynomial) Unable to be factorized into polynomials of lower degree, as (x2 + 1). (number theory, of a fraction) Whose numerator and denominator share no common factor greater than 1. (number theory, of an integer) Unable to be factored into smaller integers; prime. (algebra, of an element of a ring) Whose only divisors are units and associates. (algebraic geometry, of an algebraic variety) Inexpressible as the union of two proper algebraic subvarieties. (topology, of a manifold) Not containing a sphere of codimension 1 that is not the boundary of a ball. (group theory, of a representation) Impossible to divide further into representations of lower dimension by means of any similarity transformation. ==== Antonyms ==== reducible ==== Derived terms ==== ==== Translations ==== === Noun === irreducible (plural irreducibles) (mathematics) Such a polynomial == Spanish == === Pronunciation === IPA(key): /ireduˈθible/ [i.re.ð̞uˈθi.β̞le] (Equatorial Guinea, Spain) IPA(key): /ireduˈsible/ [i.re.ð̞uˈsi.β̞le] (Latin America, Philippines) Rhymes: -ible Syllabification: i‧rre‧du‧ci‧ble === Adjective === irreducible m or f (masculine and feminine plural irreducibles) irreducible === Further reading === “irreducible”, in Diccionario de la lengua española [Dictionary of the Spanish Language] (in Spanish), online version 23.8.1, Royal Spanish Academy [Spanish: Real Academia Española], 15 December 2025