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Let    μ  (  f  )    (respectively,    c  (  f  )   ) be the Milnor number (respectively, the degree) of the conductor of an irreducible power series    f  ∈  K  [  x  ,  y  ]   , where   K   is an algebraically closed field of characteristic    p  ⩾  0   . It is well known that    μ  (  f  )  ⩾  c  (  f  )   . We give necessary and sufficient conditions for the equality    μ  (  f  )  =  c  (  f  )    in terms of the semigroup associated with   f  , provided that    p  >  ord    f   .

The Milnor number of plane irreducible singularities in positive characteristic