On Thursday, September 17, 2015 at 2:54:15 AM UTC-7, Simon King wrote:
>
> The real problem here is that *conversion* gives rise to an error 
> that mentions *coercion*. 
>
>   sage: K.<x> = GF(25) 
>   sage: L.<y> = GF(25) 
>   sage: K(y) 
>   Traceback (most recent call last): 
>   ... 
>   TypeError: unable to coerce from a finite field other than the prime 
> subfield 
>
> That's clearly a bug. Conversion should work, even if it isn't canonical 
> and thus doesn't qualify as coercion! 
>
 
I don't think it's "clearly a bug". Conversions are *allowed* to work when 
there is a non-canonical map (i.e., a section map back that's not a 
morphism, as for ZZ -> ZZ/p ) but I don't think conversions are *required* 
to work when such a map exists.

What would the condition be? That the minimal polynomials of x,y over GF(5) 
are identical? That the minimal poly of y has a root over K and then just 
choose a root (and let the conversion system try to keep things sane)? 

Do we expect for
N.<a> = NumberField(x^5-x+1)
M.<b> = NumberField(x^5 + 5*x^4 + 8*x^3 + 4*x^2 - 1)

that

N(b)

should work? (there is actually less ambiguity there!)

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