It seems there is an issue either in the Singular interface or in Singular:

sage: K0=GF(11)
sage: #K0=QQ
sage: R0.<b>=K0[]
sage: K.<b>=K0.extension(b^5+4)
sage: R1.<zzz>=K[]
sage: L=FractionField(R1)
sage: R.<x>=L[]
sage: f=x^4+1/(b*zzz) 
sage: f._singular_()  #  where is the fraction 1/(b*zzz)  ? 
x^4
sage: f=x^4+1/(b)
sage: f._singular_()
-3*b^4+x^4
sage: f=x^4+(1)/(zzz)
sage: f._singular_()
x^4+1/(zzz)

see also 
http://ask.sagemath.org/question/25083/bug-in-roots/

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