Simon,

>> sage: K1 = GF(8,'x')
>> sage: K2 = GF(8,'y')
>> sage: K1(1) == K2(1)
>> False
>
> Not a bug, because cross-parent comparison would only make sense if
> there is some kind of a canonical map of one parent (and not just of a
> single element!) into the other or of both parents into a third one.

What I think is unreliable is that "deep down" you decide that two finite
fields are equal (or not) based on a string comparison.

If two functions create independently two GF(8,'x') then you decide that
they mean the same, and if two functions create a GF(8,'y') and GF(8,'x')
then you decide that they mean something different.

The only consequence I see to that is that we should *always* use the same
variable name for a finite field in Sage's source code, which is a very odd
rule. At least it gets the code to work.

If would feel better if two GF(8,'x') were always different, or if all
GF(8,<whatever>) were equal.

Nathann

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