Ralf Hemmecke wrote:
>
> Should SimpleAlgebraicExtension perhaps have some more words about its
> argument M. For example that M should be squarefree or irreducible?
>
> Otherwise an element might not return 0 when plugged into its minimal
> polynomial.
AFAICS SimpleAlgebraicExtension in itself should work fine if
for example M is squre, simply resulting ring will have
nilpotent elements. Of course, current definition of
'minimalPolynomial' is wrong if we allow general M.
> Even worse, S claims to be a field, but it obviously has
> zero divisors.
Yes, when R is a field SAE claim to be a field which may
cause incorrect results if M is not irreducible. So, when
R is a field M should be irreducible. I am not sure how
to best fix this situation. We have a few similar places
in algebra where declaration assert something which is
not guaranted to be true. In fact, we assert that
Complex is a field and this is exactly the same problem.
One possible solution is to document limitation. Slightly
better would be to provide two constructors, one which
builds fields, the other giving only rings. In fact,
it could be useful to allow boolean arguments in category
tests so user could supply extra argument saying that
domain is a field (or not). However final responsibility
would be on user to give correct assertions.
--
Waldek Hebisch
[email protected]
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