irreducible

Etymology

From ir- +‎ reducible.

Pronunciation

• IPA^(key): /ˌɪɹɪˈdjuːsɪbəl/

• Audio (US):

  (file)

Adjective

irreducible (comparative more irreducible, superlative most irreducible)

1. Not able to be reduced or lessened.

   Antonym: unincreasable

   • 1948 January and February, Cecil J. Allen, “British Locomotive Practice and Performance”, in Railway Magazine, page 48:

     With each reduction in the number of railways, there must come, eventually, a decline in interest, if only through reduction in variety; and when it comes to one nationalised railway only we have reached the irreducible minimum.

2. Not able to be brought to a simpler or reduced form.
3. (mathematics, of a polynomial) Unable to be factorized into polynomials of lower degree, as (x² + 1).
4. (number theory, of a fraction) Whose numerator and denominator share no common factor greater than 1.
5. (number theory, of an integer) Unable to be factored into smaller integers; prime.
6. (algebra, of an element of a ring) Whose only divisors are units and associates.
7. (algebraic geometry, of an algebraic variety) Inexpressible as the union of two proper algebraic subvarieties.
8. (topology, of a manifold) Not containing a sphere of codimension 1 that is not the boundary of a ball.
9. (group theory, of a representation) Impossible to divide further into representations of lower dimension by means of any similarity transformation.

Noun

irreducible (plural irreducibles)

1. (mathematics) Such a polynomial

Pronunciation

• IPA^(key): (Spain) /ireduˈθible/ [i.re.ð̞uˈθi.β̞le]
• IPA^(key): (Latin America, Philippines) /ireduˈsible/ [i.re.ð̞uˈsi.β̞le]
• Rhymes: -ible
• Syllabification: i‧rre‧du‧ci‧ble

Adjective

irreducible m or f (masculine and feminine plural irreducibles)

1. irreducible

Further reading

• “irreducible”, in Diccionario de la lengua española [Dictionary of the Spanish Language] (in Spanish), online version 23.8, Royal Spanish Academy [Spanish: Real Academia Española], 2024 December 10