let the category setup handle the ideals

[fchapoton]

This is removing the ideal methods in the old Ring and Field classes, moving them to the category setup.

Also removing one custom ideal method in multiple polynomial rings.

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[fchapoton]

note : the removal of "ideal" in multi-polynomial requires a careful check, as it may have hidden side-effects

[fchapoton]

At first sight, it seems to be ok thanks to the _ideal_class_ method in src/sage/rings/polynomial/multi_polynomial_ring_base.pyx

[mantepse]

It would be good to have a test for this line, in particular because IndexError was added.

[fchapoton]

so far, I was not able to find a ring that pass through that case.

[mantepse]

https://en.wikipedia.org/wiki/Euclidean_domain has $\mathbb R[x,y]/(x^2+y^2+1)$, but sage does not recognise it as principal ideal domain (or does $\mathbb Q$ vs $\mathbb R$ make a difference?):

sage: R.<x,y> = PolynomialRing(QQ)
sage: Q = R.quotient(x^2+y^2+1)
sage: Q in PrincipalIdealDomains()
False

Also, if I made no mistake, $\mathbb Q(\sqrt{19})$ would constitute an example of a principal ideal domain which is not Euclidean, but sage does not know it (or do I misunderstand the code?):

sage: R.<x> = QQ[]
sage: K.<a> = QQ.extension(x^2 + 19); K
Number Field in a with defining polynomial x^2 + 19
sage: K
Number Field in a with defining polynomial x^2 + 19
sage: K in PrincipalIdealDomains()
True
sage: K in EuclideanDomains()
True

(Disclaimer: I know nothing about this kind of math)

So, I guess it's best to leave it in.

[fchapoton]

One can declare the category a posteriori

Q._refine_category_(PrincipalIdealDomains())

but then creating the ideal follows a specific code-path for quotient rings.

For the second part, I guess you mean the ring of integers rather than the number field. I tried, and there is also a specific code-path for ideals there.

[mantepse]

I played a little with the patch and it looks good to me. I found a bug, but that's present in develop, too:

sage: I.<x> = InfinitePolynomialRing(QQ)
sage: II = I.ideal( [x[0]^3,x[1]^3+x[0]^2, x[1]^2] )
sage: x[0]^2 in II
False
sage: I.<x0, x1> = PolynomialRing(QQ)
sage: II = I.ideal( [x0^3,x1^3+x0^2, x1^2] )
sage: x0^2 in II
True

[mantepse]

LGTM

[fchapoton]

thanks ! So positive review ?

[mantepse]

sorry, forgot