base_ring of orders in relative number fields is wrong #4738

ncalexan · 2008-12-07T20:03:53Z

I think that the last ring should be the ring of integers of the relative number field.

sage: K = NumberField([x^2 + 2, x^2 + 3], 'a')
sage: K.base_ring()
Number Field in a1 with defining polynomial x^2 + 3
sage: K.maximal_order()
Relative Order in Number Field in a0 with defining polynomial x^2 + 2 over its base field
sage: K.maximal_order().base_ring()
Integer Ring

CC: @williamstein

Component: number fields

Keywords: base ring relative number field order

Stopgaps: todo

Author: Alex Ghitza

Branch/Commit: u/AlexGhitza/ticket/4738 @ 2b45932

Issue created by migration from https://trac.sagemath.org/ticket/4738

The text was updated successfully, but these errors were encountered:

ncalexan · 2008-12-07T20:04:27Z

comment:1

Sorry, the ring of integesr of the base field of the relative number field.

ncalexan · 2008-12-07T20:33:49Z

comment:2

While trying to fix this, I discovered that finding orders in relative number fields is ridiculously slow. I think pari probably will help here, either but computing it directly or by computing a basis for the absolute order and then constructing a basis for the relative order from it.

sage: K.<a,b> = NumberField([x^4 + 1, x^4 - 3])
sage: K.base_field().maximal_order()
Maximal Order in Number Field in b with defining polynomial x^4 - 3
sage: K.absolute_field('c').maximal_order()
Maximal Order in Number Field in c with defining polynomial x^16 - 8*x^12 + 432*x^8 + 640*x^4 + 256
sage: K.absolute_field('c').maximal_order().basis()
[3/512*c^14 + 3/352*c^12 + 3/128*c^10 + 3/704*c^8 + 1/32*c^6 + 15/176*c^4 + 1/44, 7/1024*c^15 + 1/512*c^14 + 3/704*c^13 + 1/256*c^12 + 3/256*c^11 + 1/128*c^10 + 3/1408*c^9 + 3/64*c^8 + 1/64*c^7 + 1/32*c^6 + 15/352*c^5 + 1/16*c^4 + 1/88*c, 35/5632*c^14 + 3/256*c^12 + 7/704*c^10 + 3/64*c^8 + 1/88*c^6 + 1/16*c^4 + 1/88*c^2, 79/11264*c^15 + 3/512*c^14 + 7/512*c^13 + 1/128*c^12 + 7/1408*c^11 + 1/128*c^10 + 3/128*c^9 + 1/32*c^8 + 1/176*c^7 + 1/32*c^6 + 1/32*c^5 + 1/176*c^3, 1/128*c^12 + 1/32*c^8 + 1/8*c^4, 1/256*c^15 + 3/256*c^13 + 1/64*c^9 + 1/16*c^5, 1/256*c^14 + 1/64*c^10 + 1/16*c^6, 1/512*c^15 + 1/128*c^13 + 1/128*c^11 + 1/32*c^7, 1/16*c^8, 1/256*c^15 + 1/32*c^9, 1/32*c^10, 1/256*c^15 + 1/128*c^13 + 1/64*c^11, 1/64*c^12, 1/64*c^13, 1/128*c^14, 1/128*c^15]
sage: K.maximal_order()
  C-c C-c---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)

/Users/ncalexan/.sage/temp/pv139196.reshsg.uci.edu/47346/_Users_ncalexan__sage_init_sage_0.py in <module>()
----> 1 
      2 
      3 
      4 
      5 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.pyc in maximal_order(self)
   4894         O = K.maximal_order()
   4895         B = [from_K(z) for z in O.basis()]
-> 4896         OK = self.order(B, check_is_integral=False, check_rank=False)
   4897         self.__maximal_order = OK
   4898         return OK

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/number_field/number_field.pyc in order(self, *gens, **kwds)
   5630             gens = gens[0]
   5631         gens = [self(x) for x in gens]
-> 5632         return order.relative_order_from_ring_generators(gens, **kwds)
   5633         
   5634 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/number_field/order.pyc in relative_order_from_ring_generators(gens, check_is_integral, check_rank, is_maximal, allow_subfield)
   1640     module_gens = [to_Kabs(a) for a in gens]
   1641     n = [a.absolute_minpoly().degree() for a in gens]
-> 1642     absolute_order_module_gens = monomials(module_gens, n)
   1643 
   1644     abs_order =  absolute_order_from_module_generators(absolute_order_module_gens,

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     15         nn = list(n)
     16         del nn[i]
---> 17         v = monomials(w, nn)
     18         k = len(v)
     19         for _ in range(n[i]-1):

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in monomials(v, n)
     52     v = Sequence(v)
     53     R = v.universe()
---> 54     return _monomials(v, R, n, 0)
     55 
     56 

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/rings/monomials.pyc in _monomials(gens, R, n, i)
     19         for _ in range(n[i]-1):
     20             for j in range(k):
---> 21                 v.append(v[j]*z)
     22             z *= gens[i]
     23         return v

/Users/ncalexan/sage-3.2.1.alpha1/local/lib/python2.5/site-packages/sage/interfaces/get_sigs.pyc in my_sigint(x, n)
      7 
      8 def my_sigint(x, n):
----> 9     raise KeyboardInterrupt
     10 
     11 def my_sigfpe(x, n):

KeyboardInterrupt:

sagetrac-fwclarke · 2009-04-21T08:41:08Z

comment:3

The problem of the slowness of computing relative maximal orders is solved
by the patch in #5842. A doctest is included at line 532 of the patched
number_field_rel.py

aghitza · 2014-04-18T01:18:39Z

Branch: u/AlexGhitza/ticket/4738

aghitza · 2014-04-18T01:19:35Z

Author: Alex Ghitza

pjbruin · 2014-04-18T10:08:51Z

Commit: 2b45932

pjbruin · 2014-04-18T10:08:51Z

comment:11

Do we really want the maximal order of the base field? I would say the base ring should be the intersection of the relative order with the base field. In the following example, the order R in L does not contain the maximal order of K:

sage: K.<s> = QuadraticField(5)
sage: O = K.order(s)  # of index 2 in the maximal order of K
sage: L.<t> = K.extension(x^2 + 1)
sage: R = L.order((s, t))
sage: a = (1 + s)/2
sage: a.is_integral()
True
sage: L(a) in R  # check that R does not contain the maximal order of K
False

Note the L(a) in the last line; this is because there is also the following bug:

sage: s in R
False  # clearly wrong because R = ZZ[s, t] as a subring of L

New commits:

`987d39f`	`fix base ring of order in relative number field`
`2b45932`	`fix base ring of orders in relative number fields`

pjbruin · 2014-04-18T10:38:19Z

comment:12

Note the L(a) in the last line; this is because there is also the following bug:
sage: s in R
False  # clearly wrong because R = ZZ[s, t] as a subring of L

This would probably be fixed by implementing a coercion morphism from the base order to the relative order. However, we currently cannot construct this morphism at all:

sage: O.hom((R(s),))
...
TypeError: images do not define a valid homomorphism

The error probably arises because Order._is_valid_homomorphism_() is not implemented.

pjbruin · 2014-05-03T15:54:36Z

comment:13

The base ring should be either Z or the order intersected with the base field (which is in general not the maximal order of the base field). We have to decide which is better.

sagetrac-jakobkroeker · 2015-08-19T22:55:32Z

Stopgaps: todo

jdemeyer · 2017-03-01T11:17:46Z

comment:17

Doctest failures.

ncalexan mannequin added this to the sage-5.11 milestone Dec 7, 2008

ncalexan mannequin added c: number theory labels Dec 7, 2008

ncalexan mannequin assigned williamstein Dec 7, 2008

ncalexan mannequin changed the title ~~base_ring of orders in relative number fields is wrong~~ base_ring of orders in relative number fields is wrong and finding maximal_orders of relative number fields is very slow Dec 7, 2008

sagetrac-fwclarke mannequin changed the title ~~base_ring of orders in relative number fields is wrong and finding maximal_orders of relative number fields is very slow~~ base_ring of orders in relative number fields is wrong Apr 21, 2009

sagetrac-mabshoff mannequin changed the title ~~base_ring of orders in relative number fields is wrong~~ base_ring of orders in relative number fields is wrong (fix at #5842) Apr 21, 2009

loefflerd mannequin added c: number fields and removed c: number theory labels Jul 21, 2009

loefflerd mannequin assigned loefflerd and unassigned williamstein Jul 21, 2009

jdemeyer modified the milestones: sage-5.11, sage-5.12 Aug 13, 2013

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.1, sage-6.2 Jan 30, 2014

aghitza added the s: needs review label Apr 18, 2014

aghitza changed the title ~~base_ring of orders in relative number fields is wrong (fix at #5842)~~ base_ring of orders in relative number fields is wrong Apr 18, 2014

pjbruin added s: needs info and removed s: needs review labels May 3, 2014

sagetrac-vbraun-spam mannequin removed this from the sage-6.2 milestone May 6, 2014

sagetrac-vbraun-spam mannequin added this to the sage-6.3 milestone May 6, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.3, sage-6.4 Aug 10, 2014

jdemeyer added s: needs work and removed s: needs info labels Mar 1, 2017

mkoeppe removed this from the sage-6.4 milestone Dec 29, 2022

sagetrac-fwclarke mannequin mentioned this issue Jun 13, 2009

Various number field improvements #5842

Closed

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

base_ring of orders in relative number fields is wrong #4738

base_ring of orders in relative number fields is wrong #4738

ncalexan mannequin commented Dec 7, 2008

ncalexan mannequin commented Dec 7, 2008

ncalexan mannequin commented Dec 7, 2008

sagetrac-fwclarke mannequin commented Apr 21, 2009

aghitza commented Apr 18, 2014

aghitza commented Apr 18, 2014

pjbruin commented Apr 18, 2014

pjbruin commented Apr 18, 2014

pjbruin commented Apr 18, 2014

pjbruin commented May 3, 2014

sagetrac-jakobkroeker mannequin commented Aug 19, 2015

jdemeyer commented Mar 1, 2017

base_ring of orders in relative number fields is wrong #4738

base_ring of orders in relative number fields is wrong #4738

Comments

ncalexan mannequin commented Dec 7, 2008

ncalexan mannequin commented Dec 7, 2008

ncalexan mannequin commented Dec 7, 2008

sagetrac-fwclarke mannequin commented Apr 21, 2009

aghitza commented Apr 18, 2014

aghitza commented Apr 18, 2014

pjbruin commented Apr 18, 2014

pjbruin commented Apr 18, 2014

pjbruin commented Apr 18, 2014

pjbruin commented May 3, 2014

sagetrac-jakobkroeker mannequin commented Aug 19, 2015

jdemeyer commented Mar 1, 2017