Remarkable is that for f = x^4+1/(b)*(1/zzz) f is correctly translated to 
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)*(1/zzz)
sage: f._singular_()  
-1/(4*zzz)*b^4+x^4

That looks problematic, but is likely a different issue from what happens 
> on the asksage question.
>

I think it is the same issue. I checked that Singular is called to compute 
the roots using Singulars factorize() method.
But even if *f* would be correctly translated to Singular the example at 
AskSage would not work,
since Singulars factorize() behaves unexpectedly in quotient rings; example 
(in Singular):

ring rng = 0,(x,b),lp; 
short = 0; 
qring qr = b^2-2; 
poly f = x^2-2; 
factorize(f); // expecting: (x-b)*(x+b) ? 
[1]: 
   _[1]=1 
   _[2]=x^2-2 
[2]: 
   1,1 

Probably I should open a ticket and move the discussion to the ticket
Am Mittwoch, 3. Dezember 2014 17:54:19 UTC+1 schrieb Nils Bruin:
>
> On Wednesday, December 3, 2014 3:07:14 AM UTC-8, Jakob Kroeker wrote:
>>
>>             ...
>> sage: f=x^4+1/(b*zzz) 
>> sage: f._singular_()  #  where is the fraction 1/(b*zzz)  ? 
>> x^4
>>
> ...
>>
> see also 
>> http://ask.sagemath.org/question/25083/bug-in-roots/
>>
>
> That looks problematic, but is likely a different issue from what happens 
> on the asksage question. I think we use libsingular for pretty much all 
> internal singular uses. We don't use the expect interface to singular for 
> communication there.
>

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