A class for ring extensions

[caruso]

Sage actually provides a rich framework for dealing with all classical algebraic structures: rings, fields, algebras, etc. Nevertheless, given (for instance) two fields K and L with K \subset L, it is not possible to build the extension L/K as a Sage object. However one can easily imagine methods related to this extension (e.g. degree, discriminant, normal_basis, decompose_on_basis, etc.)

With Bruno Grenet, Johan Rosenkilde and Luca De Feo, we raised this issue at Sage Days 75. A summary of our discussion is available ​here.

This ticket implements very generic classes for ring extensions. Other more specific classes (e.g. for field extensions or even finite field extensions) are coming soon...

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​556da4d

Coercion system improved

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b5e3eed	Coercion in scalar restriction and scalar extension
​c82968f	Small tutorial

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​f0a7ce4	Doctest from the class AlgebraFromMorphism and some methods of this class
​b6c9a33	Doctest fixed
​67c82c7	Merge branch 'develop' into 21413/class_ring_extension
​8d70dbe	Code split in several files. More doctests.
​efa3689	Doctest for BaseActionOnRing
​7f8ce2d	Doctest for the class AlgebraToRing_coercion
​991cc70	Doctest for the class AlgebraFMElement
​036fb2a	Adding licence & author

[jsrn]

Wow, this is great Xavier! Sorry for not being more active before - I promise to look more at it tomorrow.

I'm wondering whether it's a good idea to merge the "framework" before the first concrete instantiation? I.e. whether we should try to get the field-extension-algebra finished before considering this as mature? (I know you have bad experience with too large patches, but if we don't put too much stuff in the field extension as a first approximation, perhaps it's ok).

Best, Johan

[jsrn]

In the following, the extension is A = L/K with L and K rings. L.base() is B. Sometimes B is a subring of K.

Right now this AlgebraFromMorphism is mainly a thin wrapper around the L, so e.g. A elements are represented internally as elements of L, and this is also how you print them. Can you try to convince me of the value from a user's POV that he can do arithmetic on his elements inside this algebra, rather than dropping the whole Algebra idea and just letting RingExtension be some non-parent object with fancy methods on it like decompose_on_basis?

What I had in mind after SD75 is that RingExtension is an Algebra as in "a vector space equipped with a bilinear product"; in other words, when I want to see L as a vector space over K, I use RingExtension. In this world, it makes sense to have arithmetic on elements of A and they should print (and possibly be internally represented by) vectors over K; because that's the whole point of why I'm wrapping L.

In that world, AlgebraFromMorphism shouldn't have a gen method because it isn't really focused on its personality as a ring (it might have a multiplicative_generator if you like). But it should definitely have a basis method.

Other remarks that I got by looking at the code. Let's see where the above discussion takes us before doing much about the stuff below:

1. Why are you not giving self._base as argument to CommutativeAlgebra.__init__? You write in a comment that we don't want a coercion map base -> self -- I don't see why not? I guess this is related to the left/right action dichotomy you have?
2. The module doc for algebra_from_morphism is too short. It should explain what this module is and what is offered. Redundancy is not bad.
3. A method "def something(self)" should never have the doc-string "Return the something of self". That's utterly unhelpful :-)
4. from_base_ring has an extremely bad name considering that it doesn't embed an element of the base. I think we need to establish some proper nomenclature and do some commutation diagrams in the doc of this module.
5. The doc for scalar_restriction is very hard to read. Attempt to write before INPUT what this means. Could we write something like

If ``newbase`` is a sub-ring of `self.base()`, then the scalar restriction of `self` to `newbase` is the ring extension `self.ring()/newbase`. The defining morphism will be the composition of `f` and ``self.defining_morphism()`` where `f` is the coercion map from `newbase` to `self.base()`.

``newbase`` can be given as an extension, ... 

``newbase`` can be given as a morphism, ...

What is the argument for allowing extensions and morphisms as arguments to scalar_restriction? Is it really something you would normally do? Couldn't the user just do A.scalar_restriction(mymorphism.codomain()), etc.?

1. I don't like ring and base. They are too short and it's not clear which is which.
2. Speaking in parents, you let K * A = A but you let A * K = L. Won't this be a major source of confusion for users? I think your nasty example with non-commutativity at the end of RingExtension doc makes this even worse! I'm also very concerned that this might break several things in Sage since you're inheriting from CommutativeAlgebra.
3. Doc of RingExtension: can we make a commuting diagram to describe the coercion between L1/K1 and L2/K2? Something like

K1 -----> L1
^         |
|         | 
|         v
K2 -----> L2

Best, Johan

[caruso]

Replying to jsrn:

Right now this AlgebraFromMorphism is mainly a thin wrapper around the L, so e.g. A elements are represented internally as elements of L, and this is also how you print them. Can you try to convince me of the value from a user's POV that he can do arithmetic on his elements inside this algebra, rather than dropping the whole Algebra idea and just letting RingExtension be some non-parent object with fancy methods on it like decompose_on_basis?

Maybe I misunderstand your proposal but I really think that a ring extension should be a parent because we really want to take advantage of the coercion stuff. Nonetheless, the question "should it derive from CommutativeAlgebra?" is arguable and I can imagine arguments in both directions.

What I had in mind after SD75 is that RingExtension is an Algebra as in "a vector space equipped with a bilinear product"; in other words, when I want to see L as a vector space over K, I use RingExtension.

It is definitely.

In this world, it makes sense to have arithmetic on elements of A and they should print (and possibly be internally represented by) vectors over K; because that's the whole point of why I'm wrapping L.

I agree. But all of this has to be implemented in subclasses. For now, I just wrote a very general class which is supposed to deal with all algebras (possibly not free, not finite...) so it is of course difficult to implement concrete methods.

But these methods clearly exist. Here is a far-from-exhaustive list: scalar_restriction (which is already implemented), __mul__ (usual product of algebras), tensor_product, scalar_extension, cayley_differentials, krull_relative_dimension, is_free, is_finite, is_flat, is_etale, is_smooth, is_open_immersion, is_closed_immersion, is_galois, galois_group. And for elements: trace, norm.

If you insist, I can put these methods in the general class and let them raise NotImplementedError... but it would be difficult to write the doctest :-)

In that world, AlgebraFromMorphism shouldn't have a gen method because it isn't really focused on its personality as a ring (it might have a multiplicative_generator if you like).

The generator of an algebra is a well-defined mathematical notion: the algebra L/K is generated by x if $K \cup {x}$ generates L as a ring. So it clearly makes sense to have a gen method.

But it should definitely have a basis method.

Well there does exist algebras which are not free. So the basis method is clearly something we want. However it should not be inserted in such a general class but in the subclass FreeAlgebraFromMorphism or FiniteFreeAlgebraFromMorphism (which is coming soon).

Why are you not giving self._base as argument to CommutativeAlgebra.__init__? You write in a comment that we don't want a coercion map base -> self -- I don't see why not? I guess this is related to the left/right action dichotomy you have?

Suppose that we create an algebra L/K with a defining morphism phi : K -> L which is not a coercion map. If K coerces to L/K through phi and L/K coerces to L, then we would derive that K coerces to L through phi.

Such a situation really occurs in every day life (at least for people working in algebraic geometry). For instance, if K has characteristic p, these people often consider K as an algebra over itself *through the Frobenius morphism* (which is definitely not a coercion map).

The module doc for algebra_from_morphism is too short. It should explain what this module is and what is offered.

Redundancy is not bad.

A method "def something(self)" should never have the doc-string "Return the something of self". That's utterly unhelpful :-)

Ok for both. I'll try to fix this.

from_base_ring has an extremely bad name considering that it doesn't embed an element of the base. I think we need to establish some proper nomenclature and do some commutation diagrams in the doc of this module.

It is not my fault :-). This method is needed by the coercion model.

The doc for scalar_restriction is very hard to read. Attempt to write before INPUT what this means. Could we write something like

Ok. I'll do it.

What is the argument for allowing extensions and morphisms as arguments to scalar_restriction? Is it really something you would normally do? Couldn't the user just do A.scalar_restriction(mymorphism.codomain()), etc.?

Sure, it is something we normally do... just as composing ring homomorphisms.

I don't like ring and base. They are too short and it's not clear which is which.

Ok for changing. I'm also not completed satisfied with these names. Do you have some propositions?

Speaking in parents, you let K * A = A but you let A * K = L. Won't this be a major source of confusion for users? I think your nasty example with non-commutativity at the end of RingExtension doc makes this even worse!

I definitely agree that it is not perfect, but I have not found something better :-). The point is that I really want to have algebras_from_morphisms_which_are_not_coercion_maps and I think that implementing the action of scalars (through the defining morphism) is the least we can do.

I'm also very concerned that this might break several things in Sage since you're inheriting from CommutativeAlgebra.

It is one reason why I'm sure that we should inherit from CommutativeAlgebra

Doc of RingExtension: can we make a commuting diagram to describe the coercion between L1/K1 and L2/K2?

Ok.

Xavier

[jsrn]

Thanks for your careful explanations. I'm learning a lot here.

Replying to carus:

Maybe I misunderstand your proposal but I really think that a ring extension should be a parent because we really want to take advantage of the coercion stuff.

Nonetheless, the question "should it derive from CommutativeAlgebra?" is arguable and I can imagine arguments in both directions.

OK, so "coercion stuff" is an argument in favour of a Parent. I guess you mean that many objects will magically coerce to the Do-What-I-Mean behaviour on methods. But if a weird/surprising behaviour follows from forcing it into the category framework, perhaps that's an argument against it.

At least having a RingExtension being a class with lots of service methods, sort of like David's RelativeFiniteFieldExtension, there will be no arithmetic problems. You could also put norm and trace there.

In this world, it makes sense to have arithmetic on elements of A and they should print (and possibly be internally represented by) vectors over K; because that's the whole point of why I'm wrapping L.

I agree. But all of this has to be implemented in subclasses. For now, I just wrote a very general class which is supposed to deal with all algebras (possibly not free, not finite...) so it is of course difficult to implement concrete methods.

OK, that makes sense. I can see why the general class would have very weak functionality then.

But these methods clearly exist. Here is a far-from-exhaustive list: .... If you insist, I can put these methods in the general class and let them raise NotImplementedError... but it would be difficult to write the doctest :-)

Perhaps that's not a bad idea: it's nicer that any algebra will have the methods you expect and then throw NotImplementedError rather than not advertising the method at all.

The doc-test can be on a more concrete class that does implement the method. If none exist for this ticket, the doc-test will just be one demonstrating the NotImplentedError for now, with the sole purpose of doc-test coverage ;-)

The generator of an algebra is a well-defined mathematical notion: the algebra L/K is generated by x if $K \cup {x}$ generates L as a ring. So it clearly makes sense to have a gen method.

Hmm, but couldn't there be multiple generators? E.g. F[x,y]/F?

Well there does exist algebras which are not free. So the basis method is clearly something we want. However it should not be inserted in such a general class but in the subclass FreeAlgebraFromMorphism or FiniteFreeAlgebraFromMorphism (which is coming soon).

OK, argument accepted. But I think this kind of stuff reinforces that we should try to design the first concrete RingExtension, e.g. FiniteFieldExtension simultaneously with this ticket. Then we'll better be able to judge what goes where and how it should look to be useful on both sides.

Suppose that we create an algebra L/K with a defining morphism phi : K -> L which is not a coercion map. If K coerces to L/K through phi and L/K coerces to L, then we would derive that K coerces to L through phi.

Such a situation really occurs in every day life (at least for people working in algebraic geometry). For instance, if K has characteristic p, these people often consider K as an algebra over itself *through the Frobenius morphism* (which is definitely not a coercion map).

Yikes! So we must disallow coercion from K -> L/K. But we should allow A(k) and k*a, a*k, k + a, etc. for k in K and a in A. And A(a * l) is different from a * A(l). Oh man...

It seems to me that this is really dangerous territory - but perhaps unavoidable. Someone implementing algorithms for CommutativeAlgebra is going to assume that there is coercion from base to self. And that multiplication of base and self elements commute!

Perhaps CommutativeAlgebra should be called something different, and AlgebraFromMorphism should have a different base class. I've cc'ed Nicolas Thiery and Simon King in this discussion to chip in.

from_base_ring has an extremely bad name considering that it doesn't embed an element of the base. I think we need to establish some proper nomenclature and do some commutation diagrams in the doc of this module.

It is not my fault :-). This method is needed by the coercion model.

Eew, yet another symptom that we're perhaps abusing the current hierarchy.

Why doesn't GF(9) have from_base_ring? What kind of Parent requires it?

What is the argument for allowing extensions and morphisms as arguments to scalar_restriction? Is it really something you would normally do? Couldn't the user just do A.scalar_restriction(mymorphism.codomain()), etc.?

Sure, it is something we normally do... just as composing ring homomorphisms.

So it is something where established mathematical notation supports this directly? It's much more explicit to just write A.scalar_restriction(Aother.ring()), so I'd prefer not supporting A.scalar_restriction(Aother), except if a properly educated mathematician would be very surprised if this didn't exist.

I don't like ring and base. They are too short and it's not clear which is which.

Ok for changing. I'm also not completed satisfied with these names. Do you have some propositions?

We should probably first figure out what to do hierarchy-wise. But perhaps keep ring -> ring_of_element and base -> ring_of_scalars. Alternative element_ring resp. scalar_ring.

Speaking in parents, you let K * A = A but you let A * K = L. Won't this be a major source of confusion for users? I think your nasty example with non-commutativity at the end of RingExtension doc makes this even worse!

I definitely agree that it is not perfect, but I have not found something better :-). The point is that I really want to have algebras_from_morphisms_which_are_not_coercion_maps and I think that implementing the action of scalars (through the defining morphism) is the least we can do.

What do you mean "is the least we can do"?

Why is this an argument to make left and right actions different? Couldn't you implement the same left/right action, namely mapping K -> A by phi, followed by multiplication in A.

Best, Johan

[caruso]

Replying to jsrn:

Perhaps that's not a bad idea: it's nicer that any algebra will have the methods you expect and then throw NotImplementedError rather than not advertising the method at all.

OK. But, on the other hand, the list of possible methods is possibly quite long. Having a parent with 90% not implemented methods is also not that nice.

The generator of an algebra is a well-defined mathematical notion: the algebra L/K is generated by x if $K \cup {x}$ generates L as a ring. So it clearly makes sense to have a gen method.

Hmm, but couldn't there be multiple generators? E.g. F[x,y]/F?

Yes, of course.

For traditional rings, the current behaviour of gen is rather stange (to me):

    sage: R.<x,y> = QQ[]
    sage: R.gen()
    x

Is that normal?

OK, argument accepted. But I think this kind of stuff reinforces that we should try to design the first concrete RingExtension, e.g. FiniteFieldExtension simultaneously with this ticket. Then we'll better be able to judge what goes where and how it should look to be useful on both sides.

I just do not want this ticket to become too big. But I agree for implementing FiniteFieldExtension here.

Yikes! So we must disallow coercion from K -> L/K. But we should allow A(k) and k*a, a*k, k + a, etc. for k in K and a in A. And A(a * l) is different from a * A(l). Oh man...

(First, let me emphasize that this issue only occurs when the defining morphism is not a coercion map.)

Currently A(k), a*k, k + a uses coercion maps (and not defining morphism): for instance, if K coerces to L then k+a is the addition in L while if K does not coerce to L, k+a produces an error. Only k*a (implemented by a left action of K on L/K) uses the defining morphism.

I agree that it is really confusing but the same confusion appears exactly in the same way in the "theory". So I assume that people who really wants to work with algebras whose defining morphisms are not coercion maps are very aware of this source of confusion!

Why left action and not right action? Because usually scalars act on the left (for instance, we write 2x and not x2 when x lies in some vector space). I nevertheless agree that this convention is not very strong. Another option would be to implement both actions (at some point I actually did this) but it then becomes impossible to multiply an element of K by an element of L without writing explicitly the conversion.

Probably the best solution would be to have a completely different operator for the action of the base through the defining morphism. But which one? Is it a good idea to override the dot operator?

Why doesn't GF(9) have from_base_ring? What kind of Parent requires it?

I think (but I'm not quite sure) that it is used when no coercing map is set from self._base to self.

So it is something where established mathematical notation supports this directly? It's much more explicit to just write A.scalar_restriction(Aother.ring()), so I'd prefer not supporting A.scalar_restriction(Aother), except if a properly educated mathematician would be very surprised if this didn't exist.

I think that sometimes mathematicians may want to invoke A.scalar_restriction(Aother). For instance assume that we have an algebra A defined over C[X]. The fibre of A at the point x is by definition the scalar extension of A with respect to the morphism C[X] -> C mapping X to x. (It is actually also A/(X-x), but sometimes we really want to think of it as a scalar extension.)

Actually we probably prefer to use the "base_change" in that case. So maybe we can implement two methods: first scalar_extension which always uses coercion maps and second base_change which accepts all constructions.

I think that implementing the action of scalars (through the defining morphism) is the least we can do.

What do you mean "is the least we can do"?

When we want to regard L as a K-algebra, a basic thing we want to have is the action of K on L defining the algebra.

[klee]

I just found this ticket and I only have a vague idea about what this ticket does. I am curious if the present ticket would solve or at least help one solve each of the following problems (of mine)? I believe that these cases are not supported by Sage yet.

1. Suppose F is a non-prime finite field, say GF(3^2). I want to construct an extension field E over F. So E,phi=F.extension(4,map=True) gives the extension field GF(3^8)and the embedding phi. This currently only works for prime fields.

2. Suppose K is a field and R is a subring of K. Suppose K is the fraction field of R. Let e be an element of K. Then d=e.denominator(R) gives a denominator of e in R such that d*e.numerator(R) is in R.

3. Let E,F be rings. Suppose phi is a homomorphism from E to F. I want to extend phi to a homomorphism psi from the polynomial ring R=E[x] to F mapping x to c in F. Thus psi(a2*x^2+a1*x+a0)==phi(a2)*c^2+phi(a1)*c+phi(a0). Perhaps I want R.hom([c],F, base=phi) or like to work.

[jsrn]

Replying to klee:

1. Suppose F is a non-prime finite field, say GF(3^2). I want to construct an extension field E over F. So E,phi=F.extension(4,map=True) gives the extension field GF(3^8)and the embedding phi. This currently only works for prime fields.

That currently works fine, just try it.

If you want to go from F back to E for any of the elements phi(E), use phi.section().

If you want to express F as a vector space over E, things are more limited. We implemented basic support for it in sage.coding.relative_finite_field_extension. This ticket is about making that functionality much more general and thought out, and getting it into the core algebra of Sage.

2. Suppose K is a field and R is a subring of K. Suppose K is the fraction field of R. Let e be an element of K. Then d=e.denominator(R) gives a denominator of e in R such that d*e.numerator(R) is in R.

That also currently works fine. I just tested with R = GF(3^2)['x']:

sage: R = GF(3^2)['x']
sage: K = R.fraction_field()
sage: e = K.random_element()
sage: e.denominator()
(z2 + 2)*x^2 + (z2 + 1)*x
sage: e.denominator().parent()
Univariate Polynomial Ring in x over Finite Field in z2 of size 3^2
sage: e.denominator() in R
True
sage: R( e.denominator() * e )
(2*z2 + 2)*x^2 + (z2 + 2)*x + z2

3. Let E,F be rings. Suppose phi is a homomorphism from E to F. I want to extend phi to a homomorphism psi from the polynomial ring R=E[x] to F mapping x to c in F. Thus psi(a2*x^2+a1*x+a0)==phi(a2)*c^2+phi(a1)*c+phi(a0). Perhaps I want R.hom([c],F, base=phi) or like to work.

That seems more tricky to do. Note that it can easily be split into the composition of two homomorphisms:

mapCoef:  E[x]  --> F[x]
          a*x^i |-> phi(a)*x^i

eval:     F[x] --> F
             x |-> c

The eval homomorphism can be created in the current Sage, but I don't know how to conveniently create mapCoef (as a Morphism object).

Best, Johan

[klee]

Replying to jsrn:

Replying to klee:

1. Suppose F is a non-prime finite field, say GF(3^2). I want to construct an extension field E over F. So E,phi=F.extension(4,map=True) gives the extension field GF(3^8)and the embedding phi. This currently only works for prime fields.

That currently works fine, just try it.

If you want to go from F back to E for any of the elements phi(E), use phi.section().

If you want to express F as a vector space over E, things are more limited. We implemented basic support for it in sage.coding.relative_finite_field_extension. This ticket is about making that functionality much more general and thought out, and getting it into the core algebra of Sage.

Nice. Good luck!

[klee]

The following reference might be useful for the design of the class:

Lattices of Compatibly Embedded Finite Fields - WIEB BOSMA, JOHN CANNON AND ALLAN STEEL 1997

as various extensions should be compatible to each other...

[defeo]

Doctest failures:

sage -t src/sage/rings/algebra_from_morphism.py
**********************************************************************
File "src/sage/rings/algebra_from_morphism.py", line 260, in sage.rings.algebra_from_morphism.AlgebraFromMorphism._pushout_
Failed example:
    E2._pushout_(E1) is E2
Expected:
    True
Got:
    False
**********************************************************************
File "src/sage/rings/algebra_from_morphism.py", line 262, in sage.rings.algebra_from_morphism.AlgebraFromMorphism._pushout_
Failed example:
    E1._pushout_(E2) is E2
Expected:
    True
Got:
    False
**********************************************************************
File "src/sage/rings/algebra_from_morphism.py", line 265, in sage.rings.algebra_from_morphism.AlgebraFromMorphism._pushout_
Failed example:
    E1._pushout_(L2) is L2
Expected:
    True
Got:
    False
**********************************************************************
File "src/sage/rings/algebra_from_morphism.py", line 272, in sage.rings.algebra_from_morphism.AlgebraFromMorphism._pushout_
Failed example:
    E1p._pushout_(E2) is L2
Expected:
    True
Got:
    False
**********************************************************************
File "src/sage/rings/algebra_from_morphism.py", line 276, in sage.rings.algebra_from_morphism.AlgebraFromMorphism._pushout_
Failed example:
    E1p._pushout_(E2p) is L2
Expected:
    True
Got:
    False
**********************************************************************
1 item had failures:
   5 of  11 in sage.rings.algebra_from_morphism.AlgebraFromMorphism._pushout_
    [119 tests, 5 failures, 0.86 s]
----------------------------------------------------------------------
sage -t src/sage/rings/algebra_from_morphism.py  # 5 doctests failed
----------------------------------------------------------------------
Total time for all tests: 0.9 seconds
    cpu time: 0.9 seconds
    cumulative wall time: 0.9 seconds

---

New commits:

​5c1c5b6

Merge 7.6.beta2

[defeo]

Had you guys notice that there is this interface in Sage, which returns a mostly empty shell:

sage: K = GF(5^2)
sage: L = GF(5^4)
sage: E = L.algebra(K, category=Semigroups())
sage: E
Free module generated by Finite Field in z4 of size 5^4 over Finite Field in z2 of size 5^2
sage: E.an_element()
2*B[0] + 2*B[z4] + 3*B[z4^2]
sage: E.categories()

[Category of finite dimensional semigroup algebras over Finite Field in z2 of size 5^2,
 Category of semigroup algebras over Finite Field in z2 of size 5^2,
 Category of associative algebras over Finite Field in z2 of size 5^2,
 Category of rngs,
 Category of associative additive commutative additive associative additive unital distributive magmas and additive magmas,
 Category of magma algebras over Finite Field in z2 of size 5^2,
 Category of magmatic algebras with basis over Finite Field in z2 of size 5^2,
 Category of magmatic algebras over Finite Field in z2 of size 5^2,
 Category of additive commutative additive associative additive unital distributive magmas and additive magmas,
 Category of additive commutative additive associative distributive magmas and additive magmas,
 Category of additive associative distributive magmas and additive magmas,
 Category of distributive magmas and additive magmas,
 Category of magmas and additive magmas,
 Category of finite semigroups,
 Category of semigroups,
 Category of magmas,
 Category of finite dimensional modules with basis over Finite Field in z2 of size 5^2,
 Category of set algebras over Finite Field in z2 of size 5^2,
 Category of vector spaces with basis over Finite Field in z2 of size 5^2,
 Category of modules with basis over Finite Field in z2 of size 5^2,
 Category of finite dimensional modules over Finite Field in z2 of size 5^2,
 Category of vector spaces over Finite Field in z2 of size 5^2,
 Category of modules over Finite Field in z2 of size 5^2,
 Category of bimodules over Finite Field in z2 of size 5^2 on the left and Finite Field in z2 of size 5^2 on the right,
 Category of right modules over Finite Field in z2 of size 5^2,
 Category of left modules over Finite Field in z2 of size 5^2,
 Category of commutative additive groups,
 Category of additive groups,
 Category of additive inverse additive unital additive magmas,
 Category of commutative additive monoids,
 Category of additive monoids,
 Category of additive unital additive magmas,
 Category of commutative additive semigroups,
 Category of additive commutative additive magmas,
 Category of additive semigroups,
 Category of additive magmas,
 Category of finite sets,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]

[SimonKing]

Replying to defeo:

Had you guys notice that there is this interface in Sage, which returns a mostly empty shell:

In what sense would it be relevant for this ticket?

[defeo]

Hello, as you might have guessed, I'm trying to resurrect this ticket.

My biggest concern is the same as Johan's: I find k*a != a*k very confusing for two reasons:

• It is silently not commutative;
• The parent of a*k is neither A nor K.

However I think the fix is simple: explicit is better than implicit, get rid of the coercion A -> L. I find the following idiom completely reasonable:

sage: K = GF(5^2); z2 = K.gen()
sage: L = GF(5^4); z4 = L.gen()
sage: A = RingExtension(L, K, K.frobenius_endomorphism())
sage: z2 * E(z4)
4*z4^3 + 3*z4^2 + 2*z4 + 2
sage: E(z4) * z2
TypeError: ...
sage: L(E(z4)) * z2
z4^3 + 2*z4^2 + 4*z4 + 3

It could be useful if the left action was represented by an operator other than *, however there is no acceptable operator in Python (. cannot be overridden and @ (matmul) was only introduced in Python 3.5).

Or maybe just have the right action be the same as the left action, which apparently you already did.

[defeo]

Replying to SimonKing:

Replying to defeo:

Had you guys notice that there is this interface in Sage, which returns a mostly empty shell:

In what sense would it be relevant for this ticket?

At the very least, it is confusing that there is a method defined on L whose name suggests it might return RingExtension(L, K), and it returns a completely unrelated object instead.

We had already discussed of various possible APIs for this, and L.algebra_over(K) was one of those. I think Johan was in favour of this kind of API, rather than cluttering the namespace with yet another mysterious name RingExtension. It would be quite confusing to have L.algebra() and L.algebra_over().

[SimonKing]

Replying to defeo:

My biggest concern is the same as Johan's: I find k*a != a*k very confusing for two reasons:

• It is silently not commutative;

No problem, from my perspective. While "+" normally denotes a commutative operation in mathematics, "*" is generally not supposed to be commutative.

• The parent of a*k is neither A nor K.

Why would that be a problem? If a=x+2 is in ZZ[x] and x=1/2 is in QQ then a*x neither is in ZZ[x] nor in QQ.

It could be useful if the left action was represented by an operator other than *, however there is no acceptable operator in Python (. cannot be overridden and @ (matmul) was only introduced in Python 3.5).

Perhaps x>>a and a<<x?

Best regards, Simon

[SimonKing]

Replying to defeo:

At the very least, it is confusing that there is a method defined on L whose name suggests it might return RingExtension(L, K), and it returns a completely unrelated object instead.

I would not expect L.algebra(K) to return RingExtension(L,K) but to return the K-algebra with basis L whose multiplication is induced by the multiplication in L (hence L is only supposed to be a multiplicative monoid).

We had already discussed of various possible APIs for this, and L.algebra_over(K) was one of those. I think Johan was in favour of this kind of API, rather than cluttering the namespace with yet another mysterious name RingExtension.

I agree. When developing SageMath, we should avoid adding further stuff to the global namespace, whenever possible.

It would be quite confusing to have L.algebra() and L.algebra_over().

Perhaps a little variation: L.as_algebra_over(K). Or, changing perspective: L is an extension of K; so, L/K should be returned by a method of K, not by a method of L. Say, K.extension(L).

[defeo]

Replying to SimonKing:

Replying to defeo:

My biggest concern is the same as Johan's: I find k*a != a*k very confusing for two reasons:

• It is silently not commutative;

No problem, from my perspective. While "+" normally denotes a commutative operation in mathematics, "*" is generally not supposed to be commutative.

I expect it to be associative, at least. With the proposed code (k*a)*k != k*(a*k).

• The parent of a*k is neither A nor K.

Why would that be a problem? If a=x+2 is in ZZ[x] and x=1/2 is in QQ then a*x neither is in ZZ[x] nor in QQ.

I know the coercion system finds common parents, but in your example parent(a*x) == parent(x*a), and I expect the same here. I know there are exceptions even to this (e.g., matrices), but here we are talking about a commutative ring A, so I expect * to behave like its internal multiplication.

By a very common stretch of notation, I am glad to accept k*a as a shorthand for phi(k)*a, but I really don't see why a*k should be a shorthand for L(a)*L(k). If we want such exotic semantics, then we should use another symbol than *. See below

It could be useful if the left action was represented by an operator other than *, however there is no acceptable operator in Python (. cannot be overridden and @ (matmul) was only introduced in Python 3.5).

Perhaps x>>a and a<<x?

That's quite ugly, but as I said, it would be the only sensible way to support such exotic semantics.

Let's keep in mind that 90% of Sage users are going to use this to work with field extensions of finite fields or number fields. They expect k*a to mean "scalar multiplication of a by k", not to raise an exception.

The uses proposed by Xavier are certainly important, and it would be nice to support them, but I think those mathematicians who want to work with them should be ready to accept a slightly more explicit notation (an explicit conversion, or a different operator). If they really want to have k*a == a*k^p, then they should use a dedicated class such as SkewPolynomialRing.

[defeo]

Replying to SimonKing:

Replying to defeo:

At the very least, it is confusing that there is a method defined on L whose name suggests it might return RingExtension(L, K), and it returns a completely unrelated object instead.

I would not expect L.algebra(K) to return RingExtension(L,K) but to return the K-algebra with basis L whose multiplication is induced by the multiplication in L (hence L is only supposed to be a multiplicative monoid).

I frankly don't know what I expect from L.algebra(K). I would run it, and see what comes out.

Perhaps a little variation: L.as_algebra_over(K). Or, changing perspective: L is an extension of K; so, L/K should be returned by a method of K, not by a method of L. Say, K.extension(L).

We had already considered this possibility. The problem is that K.extension() is already defined for most rings, and it returns a ring. It wouldn't break compatibility, but it would be a bit surprising if QQ.extension(x^2+1) returned a number field, whereas QQ.extension(K) returned an algebra object.

Of course, it could be K.extension_algebra(L).

[mderickx]

I am just reading up on this ticket, it has quote a long discussion. Does it still really need info? If so could someone who knows what info put that in the ticket description?

[jsrn]

The "info" the ticket needs is a few decisions on trickier points of the design. But perhaps what would be most useful is a concrete suggestion by updating the ticket's code. I would personally also really like to see how e.g. a finite field extension of #20284 would be instantiated in the current - quite abstract - proposal.

[mderickx]

Yes I agree that is important to get the design right, especially since this touches something that touches quite a few areas of sage, since finite fields are far from the only place that one wants to consider relative extensions.

It is quite difficult for me to grasp what the "current proposal" is from the all the discussion that has been going on here. Or is the the ​one linked to in the description. I need this for something else as well so I plan to work on this.

What I think is that sage already has a class for relative extensions, and that class is called CommutativeAlgebra? from sage.rings.ring . The only problem is that right now that class is a dummy class which has almost nothing implemented yet.

So what I propose is the following, try to put as much general code as possible in CommutativeAlgebra? and set up a framework there that makes it easy later for concrete implementations of rings/fields to see themselves as Algebras over over other bases then their current base. And use this framework to make things work as follows:

sage: GF(81).vector_space() #note this already works
Vector space of dimension 4 over Finite Field of size 3
sage: GF81_as_GF_9_algeba = Algebras(GF(9))(GF(81))
sage: GF81_as_GF_9_algeba.vector_space()
Vector space of dimension 2 over Finite Field in z2 of size 3^2

For the more complicated cases where there is no coercion morphism yet we can try to make the following work

Algebras(GF(9))(GF(81), morphism=my_morhpism)

[defeo]

The main objection by @caruso, and the reason for the code he wrote, was that most on the interface, and some algorithms, can also be useful for non-commutative algebras.

He is especially interested in "skew algebras". Citing @caruso:

The point is that I really want to have algebras_from_morphisms_which_are_not_coercion_maps and I think that implementing the action of scalars (through the defining morphism) is the least we can do.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​656ec77

Small fixes

[jsrn]

I inexplicably got remotivated to revive and work on this, despite its marathon discussion and open questions. I'll try to summarise the design problems we are facing along with solution proposals in the following.

Naming: Of the now many suggestions, I really like that the algebra L/K could be obtained from K.extension_algebra(L). AlgebraWithMorphism should not be exposed in the global namespace since it is too abstract to have any useful functionality in itself.

The commutativity issue: The issue is that in the current proposal, if A = L/K uses a defining map phi which is _not_ the coercion map of K in L, then k*a == phi(k)*a, while a*k == a*L(k). This could cause confusion for users, but also developers since the class of A currently derives from CommutativeAlgebra (In A we do guarantee that a1 * a2 = a2 * a1)

It has been suggested to let * be commutative and mean k * a = a * k = a * L(k), and introduce a new operator for phi(k) * a. I propose using the binary-and operator for this purpose, &, and hence

   k & a = a & k = phi(k) * a = a * phi(k) = A(k) * a = a * A(k)

The issue with L.algebra: The issue is that there is an existing method

L.algebra(K), which does something different from what we want. One way to alleviate confusion suggested by Simon is to introduce the method L.as_algebra_over(K). I see no reason why we could not have both this as well as K.extension_algebra(L).

Writing GF(q^m)/GF(q) in this ticket: I still advocate that we should write a

(bare-bone) implementation of this particular ring extension in this ticket. We need to see the abstract class in action to properly judge whether the design is at least not crazy.

[klee]

Replying to jsrn:

Writing GF(q^m)/GF(q) in this ticket: I still advocate that we should write a

(bare-bone) implementation of this particular ring extension in this ticket. We need to see the abstract class in action to properly judge whether the design is at least not crazy.

Actually I have a plan to remove the experimental RelativeFiniteFieldExtension module from the coding section of Sage, right after the cleanup ticket #27634. For one thing, the module is ill-placed. For another, codes can live without it -- I checked this.

By the way, welcome back Johan!

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ee0e074

Rewrite documentation and fix bugs

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​a785d8d

Fix bug in coercion

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​77332e0	Disable the construction L/K
​631f228	backend morphism and tower of extensions

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​07481dd

Ring extensions with basis

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​22d7f4d

add class RingExtensionWithGen

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2566db5

Make things relative

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​6fc7e5f

move code and rename classes

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​81e4a22

check arguments for finite free ring extensions

[saraedum]

is_finite should not return True btw as this is about cardinality.

[saraedum]

It appears that your homs cannot handle generators?

The following happens in the new finite field code:

sage: GF(3^2).extension(3, absolute=False)._any_embedding(GF(3^12))

produces

        if codomain.has_coerce_map_from(self):
      File "sage/structure/parent.pyx", line 1969, in sage.structure.parent.Parent.has_coerce_map_from (build/cythonized/sage/structure/parent.c:16392)
        cpdef bint has_coerce_map_from(self, S) except -2:
      File "sage/structure/parent.pyx", line 1991, in sage.structure.parent.Parent.has_coerce_map_from (build/cythonized/sage/structure/parent.c:16343)
        return self._internal_coerce_map_from(S) is not None
      File "sage/structure/parent.pyx", line 2133, in sage.structure.parent.Parent._internal_coerce_map_from (build/cythonized/sage/structure/parent.c:17274)
        mor = self.discover_coerce_map_from(S)
      File "sage/structure/parent.pyx", line 2270, in sage.structure.parent.Parent.discover_coerce_map_from (build/cythonized/sage/structure/parent.c:17722)
        user_provided_mor = self._coerce_map_from_(S)
      File "sage/rings/finite_rings/finite_field_base.pyx", line 1334, in sage.rings.finite_rings.finite_field_base.FiniteField._coerce_map_from_ (build/cythonized/sage/rings/finite_rings/finite_field_base.c:15164)
        return R.hom((self.gen() ** ((self.order() - 1)//(R.order() - 1)),))
      File "sage/structure/parent_gens.pyx", line 291, in sage.structure.parent_gens.ParentWithGens.hom (build/cythonized/sage/structure/parent_gens.c:3689)
        return parent.Parent.hom(self, im_gens, codomain, base_map=base_map, category=category, check=check)
      File "sage/structure/parent.pyx", line 1363, in sage.structure.parent.Parent.hom (build/cythonized/sage/structure/parent.c:11767)
        return self.Hom(codomain, **Hom_kwds)(im_gens, **kwds)
      File "/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/rings/ring_extension_homset.py", line 7, in __call__
        return RingExtensionHomomorphism(self, *args, **kwargs)
      File "sage/rings/ring_extension_morphism.pyx", line 74, in sage.rings.ring_extension_morphism.RingExtensionHomomorphism.__init__ (build/cythonized/sage/rings/ring_extension_morphism.c:4992)
        backend = backend_morphism(backend)
      File "sage/rings/ring_extension_morphism.pyx", line 33, in sage.rings.ring_extension_morphism.backend_morphism (build/cythonized/sage/rings/ring_extension_morphism.c:3757)
        g = _backend_morphism(f)
      File "sage/rings/ring_extension_morphism.pyx", line 22, in sage.rings.ring_extension_morphism._backend_morphism (build/cythonized/sage/rings/ring_extension_morphism.c:3355)
        if not isinstance(f.domain(), RingExtension_class) and not isinstance(f.codomain(), RingExtension_class):
    AttributeError: 'list' object has no attribute 'domain'

where any_embedding looks like this:

    def _any_embedding(self, codomain):
        if codomain.has_coerce_map_from(self):
            return codomain.coerce_map_from(self)

        base_hom = self.base_ring()._any_embedding(codomain)
        minpoly = self.gen().minpoly().change_ring(base_hom)
        return self.hom(codomain, [minpoly.any_root()], base_map=base_hom)

Here is some more info about this:

sage: k = GF(3^2).extension(3, absolute=False)
sage: l = GF(3^12)
sage: k._defining_morphism
Ring morphism:
  From: Finite Field in z2 of size 3^2
  To:   Finite Field in b729 of size 3^6
  Defn: z2 |--> 2*b729^5 + 2*b729^3 + b729^2 + 2*b729 + 2
sage: l.has_coerce_map_from(k) *boom*

[saraedum]

Also, it seems that you want to make this change to ignore any check here

diff --git a/src/sage/rings/ring_extension_morphism.pyx b/src/sage/rings/ring_extension_morphism.pyx
index 3028d763ec..644273ddd3 100644
--- a/src/sage/rings/ring_extension_morphism.pyx
+++ b/src/sage/rings/ring_extension_morphism.pyx
@@ -63,7 +63,7 @@ cdef class RingExtensionHomomorphism(RingHomomorphism):
         sage: E2 = RingExtension(L,K)
 
     """
-    def __init__(self, parent, backend):
+    def __init__(self, parent, backend, check=None):
         RingHomomorphism.__init__(self, parent)
         backend_domain = self.domain()
         if isinstance(backend_domain, RingExtension_class):

[saraedum]

I think that RingExtension_class shouldn't be a UniqueRepresentation. Instead RingExtension should be a factory. Otherwise, I probably can't inherit from RingExtension_class for classes that are served through a UniqueFactory.

[saraedum]

Working on p-adic extensions with this I am now stuck at this that I cannot make much sense of.

Go to ​https://gitlab-hooks-flau3jeaza-ew.a.run.app/status/dev/branch/general-extensions click "Launch Binder" and type in

L.<a> = Qp(2).extension(x)

to get

/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/rings/ring_extension.pyc in __init__(self, defining_morphism, basis, names, coerce, check)
    616     def __init__(self, defining_morphism, basis, names=None, coerce=False, check=True):
    617         RingExtension_class.__init__(self, defining_morphism, coerce)
--> 618         self._basis = [ self(b) for b in basis ]
    619         if names is None:
    620             names = [ ]

/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/structure/parent.pyx in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9203)()
    898         if mor is not None:
    899             if no_extra_args:
--> 900                 return mor._call_(x)
    901             else:
    902                 return mor._call_with_args(x, args, kwds)

/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/rings/ring_extension_morphism.pyx in sage.rings.ring_extension_morphism.RingExtensionHomomorphism._call_ (build/cythonized/sage/rings/ring_extension_morphism.c:5278)()
     85         if isinstance(self.domain(), RingExtension_class):
     86             x = x._backend()
---> 87         y = self._backend_morphism(x)
     88         if isinstance(self.codomain(), RingExtension_class):
     89             y = self._codomain(y)

/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/categories/map.pyx in sage.categories.map.Map.__call__ (build/cythonized/sage/categories/map.c:6517)()
    771         if P is D: # we certainly want to call _call_/with_args
    772             if not args and not kwds:
--> 773                 return self._call_(x)
    774             return self._call_with_args(x, args, kwds)
    775         # Is there coercion?

/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/rings/morphism.pyx in sage.rings.morphism.RingHomomorphism_im_gens._call_ (build/cythonized/sage/rings/morphism.c:9428)()
   1285             2*x + 3*y + 2*z
   1286         """
-> 1287         return x._im_gens_(self.codomain(), self.im_gens(), base_map=self.base_map())
   1288 
   1289 

/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element._im_gens_ (build/cythonized/sage/structure/element.c:5592)()
    615         tuple of elements of im_gens.
    616         """
--> 617         raise NotImplementedError
    618 
    619     cpdef base_extend(self, R):

NotImplementedError: 

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c99fa82	morphisms between RingExtensions now support im_gens
​d4a35ac	Add base_map to homomorphisms defined by images of generators
​1f78dd4	Fix some doctests
​84704ba	Fix doctests and py3, incorporate small reviewer suggestion
​6a52519	Merge branch 't/26105/26105_base_hom' into t/21413/21413/class_ring_extension
​0433a1a	more work on morphisms

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e933a5d

write a factory

[saraedum]

I think we should change this:

/projects/ab5d7b52-0433-4883-9a36-a20aa7103d52/sage/local/lib/python2.7/site-packages/sage/rings/ring_extension.py(699)__init__()
    697             RingExtensionWithBasis.__init__(self, defining_morphism, basis, names, coerce, check)
    698         except ValueError:
--> 699             raise ValueError("the given element is not a generator")

The actual error was: *** ValueError: too many values to unpack. Essentially a syntax error that now manifested in that strange way.

We probably do not want to catch at all here but instead add a comment in the line above RingExtensionWithBasis.__init__(…) that says something like if you see a ValueError here, gen is probably not a generator.

[saraedum]

You might want to cherry pick these two commits:

​https://gitlab.com/sagemath/sage/merge_requests/33/diffs?commit_id=2482826f13425776a95576fbe8a602b636259ad3

​https://gitlab.com/sagemath/sage/merge_requests/33/diffs?commit_id=d044c89f78c0d0b6f6db2042b5c1e30ddb8b27b8

(or just manually copy them over.)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ed7c7b6

fix variable_name(s)

[caruso]

We worked with David Roe and Julian Rüth on this ticket this week and finally decided to follow a different approach. We defined a coercion map from K to the extension A = L/K (coming from the defining morphism) but no coercion map from A to L (except when K coerces to L through the defining morphism of the extension). Consequently a*k is always the same as k*a (and is equal to phi(k)*a where phi is the defining morphism) but differs from l*k.

This point of view looked good to us because, in the use cases we have in mind (typically implementing field towers), the defining morphisms are basically never coercion maps but we definitely don't want to use the @ operator to perform multiplications in the tower.

We also implemented special classes for finite field extensions. Below is a short demo showing the capabilities provided by this ticket.

We first create several a bunch of number fields and define a tower of extension QQ -> AA -> BB -> CC. (Currently, the tower is not that easy to define because we still miss a good extension method; we will implement it soon, hopefully.)

sage: A.<a> = QQ.extension(x^2 - 2)
sage: B.<b> = QQ.extension(x^4 - 2)
sage: C.<c> = QQ.extension(x^12 - 2)

# We create extensions
sage: AA.<a> = RingExtension(QQ)
sage: f = Hom(AA,B)(A.hom([b^2]))
sage: BB.<b> = RingExtension(f)
sage: g = Hom(BB,C)(B.hom([c^3]))
sage: CC.<c> = RingExtension(g)

Now we have our tower QQ -> AA -> BB -> CC. Let's play with it.

sage: AA
Ring in a with defining polynomial x^2 - 2 over its base
sage: BB
Ring in b with defining polynomial x^2 - a over its base
sage: CC
Ring in c with defining polynomial x^3 - b over its base

sage: CC.base_field()
Ring in b with defining polynomial x^2 - a over its base
sage: CC.base_field() is BB
True
sage: BB.base_field() is AA
True
sage: CC.intermediate_rings()
[Ring in c with defining polynomial x^3 - b over its base,
 Ring in b with defining polynomial x^2 - a over its base,
 Ring in a with defining polynomial x^2 - 2 over its base,
 Number Field in void with defining polynomial x]

sage: CC.gens()
(c,)

The gens method has also a relative version

sage: CC.gens(AA)   # generators over AA
(c,b)
sage: CC.gens(QQ)
(c,b,a)

And also:

sage: CC.basis()
[1, c, c^2]
sage: CC.basis(AA)
[1, c, c^2, b, b*c, b*c^2]
sage: CC.basis(QQ)
[1, c, c^2, b, b*c, b*c^2, a, a*c, a*c^2, a*b, a*b*c, a*b*c^2]

Now let's play with elements

sage: u = CC.random_element(); u
((-1/2 + 2*a) - 31*b) + (2 + (1 + 1/2*a)*b)*c + ((5/2 - 1/2*a) + 3*b)*c^2
sage: u.trace()     # trace over the base field BB
(-3/2 + 6*a) - 93*b
sage: u.norm()
(95447/8 - 5353/8*a) + (-2443/4 - 115345/4*a)*b
sage: u.matrix()
[  (-1/2 + 2*a) - 31*b     2 + (1 + 1/2*a)*b   (5/2 - 1/2*a) + 3*b]
[3*a + (5/2 - 1/2*a)*b   (-1/2 + 2*a) - 31*b     2 + (1 + 1/2*a)*b]
[        (1 + a) + 2*b 3*a + (5/2 - 1/2*a)*b   (-1/2 + 2*a) - 31*b]
sage: u.polynomial()
((5/2 - 1/2*a) + 3*b)*x^2 + (2 + (1 + 1/2*a)*b)*x + (-1/2 + 2*a) - 31*b
sage: u.polynomial()(c) == u
True

Similarly, all the above functions have relative versions

sage: u.trace(QQ)
-6
sage: u.trace().trace().trace() == u.trace(QQ)
True
sage: u.matrix(AA)
[ -1/2 + 2*a           2 5/2 - 1/2*a         -31   1 + 1/2*a           3]
[        3*a  -1/2 + 2*a           2 5/2 - 1/2*a         -31   1 + 1/2*a]
[      1 + a         3*a  -1/2 + 2*a           2 5/2 - 1/2*a         -31]
[      -31*a       1 + a         3*a  -1/2 + 2*a           2 5/2 - 1/2*a]
[ -1 + 5/2*a       -31*a       1 + a         3*a  -1/2 + 2*a           2]
[        2*a  -1 + 5/2*a       -31*a       1 + a         3*a  -1/2 + 2*a]
sage: P = u.polynomial(QQ); P
3*x0^2*x1 - 1/2*x0^2*x2 + 1/2*x0*x1*x2 + 5/2*x0^2 + x0*x1 + 2*x0 - 31*x1 + 2*x2 - 1/2
sage: P(c,b,a) == u   # (c,b,a) are the generators of CC over QQ
True

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​7d88f9a	Working on p-adic free module isomorphisms
​f4f17d6	Fix precision problem in vector space maps by using new method _polynomial_list
​23947e9	Working on vector_space and free_module
​8db48a2	Reviewer suggestions
​79f2458	One more Returns
​9313b2c	Adjust comment
​ad98d9c	Address reviewer comments
​8615a3d	Merge branch 't/28481/free_module' into t/21413/21413/class_ring_extension
​9faeb90	Fix small bugs
​f625e62	make free_module in number fields cooler

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8dcf149

implement fraction_field and fix division