The Collaborative International Dictionary of English v.0.48:

Irreducible \Ir`re*du"ci*ble\, a.
   1. Incapable of being reduced, or brought into a different
      state; incapable of restoration to its proper or normal
      condition; as, an irreducible hernia.
      [1913 Webster]

   2. (Math.) Incapable of being reduced to a simpler form of
      expression; as, an irreducible formula.
      [1913 Webster]

   Irreducible case (Alg.), a particular case in the solution
      of a cubic equation, in which the formula commonly
      employed contains an imaginary quantity, and therefore
      fails in its application. -- Ir`re*du"ci*ble*ness, n. --
      -- Ir`re*du"ci*bly, adv.
      [1913 Webster]

WordNet (r) 3.0 (2006):

irreducible
    adj 1: incapable of being made smaller or simpler; "an
           irreducible minimum"; "an irreducible formula"; "an
           irreducible hernia" [ant: reducible]

Moby Thesaurus II by Grady Ward, 1.0:

90 Moby Thesaurus words for "irreducible":
   a certain, an, any, any one, atomic, austere, bare, basic,
   called for, chaste, deep-seated, either, elementary, esoteric,
   essential, exclusive, fundamental, homely, homespun, homogeneous,
   immanent, implanted, implicit, inalienable, indicated,
   indispensable, individual, indivisible, indwelling, inextricable,
   infixed, ingrained, inherent, inner, integral, internal, intrinsic,
   inward, inwrought, irreductible, irreplaceable, lone, mere,
   monadic, monistic, monolithic, necessary, needed, needful,
   of a piece, one, plain, prerequisite, primal, primary, private,
   pure, pure and simple, required, requisite, resident, secret,
   severe, simon-pure, simple, single, singular, sole, solid,
   solitary, spare, stark, subjective, unadorned, unalienable,
   unanalyzable, unchallengeable, uncluttered, undifferenced,
   undifferentiated, undivided, unforgoable, uniform, unique, unitary,
   unquestionable, unsolvable, vital, wanted, whole