Opened 12 years ago
Closed 11 years ago
#9334 closed enhancement (fixed)
Implement Hilbert symbols over number fields
Reported by: | Alyson Deines | Owned by: | David Loeffler |
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Priority: | major | Milestone: | sage-4.8 |
Component: | number fields | Keywords: | hilbert symbol |
Cc: | Marco Streng, Jeroen Demeyer | Merged in: | sage-4.8.alpha1 |
Authors: | Aly Deines, Marco Streng, Jeroen Demeyer | Reviewers: | David Loeffler, John Cremona, Marco Streng, Jeroen Demeyer |
Report Upstream: | Fixed upstream, in a later stable release. | Work issues: | |
Branch: | Commit: | ||
Dependencies: | #11304, #11540, #11130 | Stopgaps: |
Description (last modified by )
PARI has Hilbert symbols for number fields. Hilbert symbols can be implemented by wrapping PARI's function nfhilbert
.
Apply trac_9334_nfhilbert.patch and 9334_review_jdemeyer.patch.
Attachments (3)
Change History (69)
comment:1 Changed 12 years ago by
Status: | new → needs_work |
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comment:2 Changed 12 years ago by
Status: | needs_work → needs_review |
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I changed the code as Tim (correctly) suggested so as it doesn't assume reduced input.
comment:4 Changed 12 years ago by
Status: | needs_review → needs_work |
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Work issues: | → patch does not apply |
This doesn't seem to apply to 4.4.4. Does it require some other patch as a prerequisite? Also, the docstrings don't seem to be correctly ReST formatted (you should always run sage -docbuild reference html
and check that there are no warnings before submitting a patch).
comment:5 Changed 12 years ago by
Work issues: | patch does not apply → ReST formatting issues |
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I see. So it's supposed to be applied on top of the patches at #9317. That's fine, but you should explain this in your trac upload messages. Don't repost random patches from other tickets on this ticket -- that's just unnecessary duplication, and it's confusing for the release maintainer when s/he has to merge stuff later.
Anyway: with the #9317 patches in place these four patches apply fine, and all doctests pass. But they're quite hard to review, since you seem to have added code in one place in the first patch and then removed it and added it again somewhere else in the second. Could I suggest that you use the Mercurial "qfold" command to combine the four patches into one single patch? That would make the reviewer's job vastly easier. And don't forget those docstring formatting problems; the two that stand out most at a quick glance are that the LaTeX formulae should be in backticks not dollar signs (`x^2 + 2`
etc), and the LaTeX fraction command is \frac
not \frak
.
comment:6 Changed 12 years ago by
Here is one single patch. It does not depend on ticket #9317. This should have all the documentation fixed. Thank you for your patience, I've learned a lot about Mercurial and ReST formatting recently.
I also applied this on a clean clone of 4.4.2 to check that it would build, all the doctest pass, and the -docbuild looks correct.
comment:7 Changed 12 years ago by
Most of this looks fine, and the docstring formatting is much better; but there are some technical issues.
- The code in
solver_mod_p
is obviously wrong for n > 1: it calculates the inverse modulo P^{n} but then takes the square root of this modulo P. You need some kind of Hensel lifting or suchlike to get an answer that's right modulo P^{n}.
- The code in
uniformizer
is a mess (e.g. it trivially fails for any non-principal ideal in a number field of degree > 2, because you've assumedself.integral_basis()
has length 2). But there's already a methodsage.rings.number_field.number_field.NumberField_generic.uniformizer
(taking a prime as an argument). I agree that it is worth having uniformizers accessible via a method of ideals as well, but it should just be a thin wrapper around the existing code.
comment:8 Changed 12 years ago by
Cc: | Marco Streng added |
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Milestone: | sage-wishlist → sage-5.0 |
Reviewers: | → David Loeffler, John Cremona |
Work issues: | ReST formatting issues → ReST formatting issues, and more |
I generally agree with David's points. This code will be very useful for a topic begin done at SD23 (solving conics over various fields) so I am keen to get this in (suitably modified).
In generalized_legendre_symbol: (1) test P for primality first, before trying to construct its residue field. (2) instead of K(2).valuation(P) just test that n is odd. (3) don't raise run-time errors, make them ValueErrors??. (4) make the return types consistent: you return either +1 in k or -1 as a python int. I would return a Sage integer in either case. (5) you do not test if P divides self. If so, return 0 (as a Sage integer)>
Why are generalized_hilbert_symbol and _legendre_symbol in sage/rings/arith.py? I would put them both in number_fields -- where you put the even one in fact.
In generalized_even_hilbert_symbol you define but do not use iprime, so delete it. And do the simple calculation to get the coefficients of jprime2 so you don't need to construct the quaternion algebra. (You can leave in a comment about that).
_voight_alg_6_2 has some ^{ symbols which should be . Check for others. }
Do what David said about uniformizer -- just call the existing function.
Sort out the solve function.
comment:9 Changed 12 years ago by
Great, I could use this.
While you are still at it, I have a small wish list as well. Could you
- Let the generalized even Hilbert symbol accept fractions (as the odd one and the QQ one do)?
sage: hilbert_symbol(1/3, 1, 2) 1 sage: K.<i> = QuadraticField(-1) sage: O = K.maximal_order() sage: generalized_hilbert_symbol(K(1/2), K(1), 3*O) 1 sage: generalized_hilbert_symbol(K(1/3), K(1), (1+i)*O) NotImplementedError: inverse_mod is not implemented for non-integral elements
- Also add the Hilbert symbol for infinite places? See e.g.
sage: hilbert_symbol(-1, -1, -1) -1
This is almost trivial compared to what you've already done. I have code, contact me if you have questions.
- Correct the doc text. The doc of generalized_even_hilbert_symbol should say that P must divide 2, while generalized_hilbert_symbol should not say that P must be odd
comment:10 Changed 12 years ago by
In addition to the first part of my precious comment: generalized_even_hilbert_symbol should accept a and b to both be divisible by p.
sage: hilbert_symbol(2,2,2) 1 sage: K.<i>=QuadraticField(-1) sage: O=K.maximal_order() sage: generalized_hilbert_symbol(3,3,3*O) 1 sage: generalized_hilbert_symbol(2,2,2*O) ValueError: P must be a prime
comment:11 Changed 12 years ago by
Oops, the last two lines of my previous comment should of course read
sage: p = 1+i sage: generalized_hilbert_symbol(p,p,p*O) RuntimeError: ord_P(a) or ord_P(b) must be zero
comment:12 Changed 12 years ago by
Priority: | minor → major |
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comment:13 Changed 12 years ago by
Description: | modified (diff) |
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Summary: | hilbert symbols!!! → Implement Hilbert symbols over number fields |
Alyson, are you intending to fix the various points raised by reviewers here? If not, someone else should. Ticket #9320 is waiting on this one.
comment:14 Changed 12 years ago by
Here are the changes I've made so far:
- in generalized_legendred_symbol I test for primality first
- instead of K(2).valuation(P) I just test that n is odd
- I've changed RunTime? Errors to ValueErrors?
- I return +/- 1 as sage integers
- I test if P|self and if so return 0 (as a sage integer)
- in generalized_hilbert_symbol I deleted iprime
- I did the calculation and have hard coded jprime^{2 }
- I've replace ^{ with where necessary in _voight_alg_6_2 }
- Things should work for fractions
- generalized_hilbert_symbol should also accept a,b, divisible by p
One question I have, does anyone know about hensel lifting in sage?
comment:15 Changed 12 years ago by
Excellent -- should it be "needs review" again then?
There must be places in Sage where Hensel lifting is done, but I do not know of any general framework for it. You could try asking David Roe, who (I think) wrote a lot of the p-adic code.
comment:16 Changed 12 years ago by
No, it shouldn't be "needs review" yet. I still need to fix solver_mod_p.
comment:17 Changed 12 years ago by
Status: | needs_work → needs_review |
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I have fixed solver_mod so that it computes the square root mod P^{n. }
comment:18 Changed 12 years ago by
Fixes/added documentation. Has the examples above in the documentation.
comment:19 Changed 12 years ago by
Status: | needs_review → needs_work |
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Based on 4.5.3.alpha1:
sage -t sage/rings/number_field/number_field_ideal.py ********************************************************************** File "/home/rlmill/sage-4.5.3.alpha1/devel/sage-main/sage/rings/number_field/number_field_ideal.py", line 1212: sage: P.uniformizer() Expected: a + 4 Got: -2*a + 1 ********************************************************************** File "/home/rlmill/sage-4.5.3.alpha1/devel/sage-main/sage/rings/number_field/number_field_ideal.py", line 1219: sage: P.uniformizer() Expected: -7*a^4 + 13*a^3 - 13*a^2 - 2*a + 50 Got: a^4 - a^3 + a^2 - a + 1 **********************************************************************
comment:20 Changed 12 years ago by
Status: | needs_work → needs_review |
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Ok, I checked at P.uniformizer() is just wrapping K.uniformizer(P). So assuming K.uniformizer(P) is correct (which it appears to be), I've fixed the documentation for P.uniformizer().
comment:21 Changed 12 years ago by
Status: | needs_review → needs_work |
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Here are some comments have already been mentioned above:
- why aren't generalized_hilbert_symbol and generalized_even_hilbert_symbol in the same file? (e.g. both in number_field as John suggested)
- the documentation of generalized_hilbert_symbol says that the prime should be odd, which isn't necessary, in fact, it would be good to have an even example in the doctest so this functionality doesn't get broken
- the documentation of generalized_even_hilbert_symbol doesn't say that the prime must be even, which it should!
Also, generalized_even_hilbert_symbol is less powerful than the general one:
sage: K.<i> = QuadraticField(-1) sage: O = K.maximal_order() sage: generalized_hilbert_symbol(K(1/3), K(1), (1+i)*O) 1 sage: generalized_even_hilbert_symbol(K(1/3), K(1), (1+i)*O) ... NotImplementedError: inverse_mod is not implemented for non-integral elements
So I guess the documentation of generalized_even_hilbert_symbol should say that the input should consist of integral elements? Possibly the documentation of generalized_even_hilbert_symbol could say that this is simply an auxiliary function and the user should call generalized_hilbert_symbol instead?
needs_work because of the documentation issues for generalized_even_hilbert_symbol
comment:22 Changed 12 years ago by
Reviewers: | David Loeffler, John Cremona → David Loeffler, John Cremona, Marco Streng |
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here's a non-documentation reason for needs_work
:
sage: K.<i> = QuadraticField(-1) sage: O = K.maximal_order() sage: generalized_hilbert_symbol(K(-1/3), K(-2/3), (1+i)*O) ... ValueError: self is not a square root mod P^n
comment:23 Changed 12 years ago by
And here's another one. In Magma, I get:
> L := QuadraticField(5); > O := MaximalOrder(L); > HilbertSymbol(L!-3, L!-2, 2*O); 1 1/2 + 1/2*i j
In pari, I get:
? k = nfinit(x^2-5) ... ? nfhilbert(k, -3, -2, idealprimedec(k, 2)[1]) %2 = 1
But the patch gives:
sage: L.<a> = QuadraticField(5) sage: generalized_hilbert_symbol(L(-3), L(-2), 2*O) -1
Something is wrong...
comment:24 Changed 12 years ago by
In fact, what are the reasons for implementing this independently rather than using pari's function?
Changed 12 years ago by
Attachment: | trac_9334_nfhilbert.patch added |
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interface to pari's nfhilbert, apply only this file
comment:25 Changed 12 years ago by
Authors: | aly.deines → Aly Deines, Marco Streng |
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Description: | modified (diff) |
Work issues: | ReST formatting issues, and more → bug in Pari 2.4.3 |
All that needed to be done was wrap pari's nfhilbert function and copy Aly's doctests, but...
One of those doctests revealed yet another bug introduced by the pari upgrade. See http://pari.math.u-bordeaux.fr/cgi-bin/bugreport.cgi?bug=1147
Apply trac_9334_nfhilbert.patch once pari bug 1147 is fixed and that fix reaches Sage.
Possible future improvements:
- implement also for relative number fields by simply delegating to the absolute case
- implement also for QQ by wrapping the global function hilbert_symbol as a member of QQ
comment:26 Changed 12 years ago by
Report Upstream: | N/A → Reported upstream. Little or no feedback. |
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comment:27 Changed 12 years ago by
Report Upstream: | Reported upstream. Little or no feedback. → Reported upstream. Developers acknowledge bug. |
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comment:28 Changed 12 years ago by
Work issues: | bug in Pari 2.4.3 → either wait for Pari bug 1147 to be fixed, or finish Aly Deines' independent Sage implementation |
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comment:29 follow-up: 30 Changed 12 years ago by
Cc: | Jeroen Demeyer added |
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Report Upstream: | Reported upstream. Developers acknowledge bug. → Fixed upstream, but not in a stable release. |
Work issues: | either wait for Pari bug 1147 to be fixed, or finish Aly Deines' independent Sage implementation → update pari in sage, and make sure all doctests in this patch are correct (or finish Aly Deines' independent implementation) |
The pari bug resulting in incorrect outputs of nfhilbert was fixed in Pari svn revision 13063 2011-04-05 14:03:38 +0100 (Tue, 05 Apr 2011).
How do we get this fix into Sage?
comment:30 Changed 12 years ago by
Replying to mstreng:
The pari bug resulting in incorrect outputs of nfhilbert was fixed in Pari svn revision 13063 2011-04-05 14:03:38 +0100 (Tue, 05 Apr 2011).
How do we get this fix into Sage?
I am currently maintaining the PARI spkg for Sage, so asking me is the best way.
comment:32 Changed 11 years ago by
Dependencies: | → #11304, #11540, #11130 |
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Description: | modified (diff) |
Report Upstream: | Fixed upstream, but not in a stable release. → Fixed upstream, in a later stable release. |
Work issues: | update pari in sage, and make sure all doctests in this patch are correct (or finish Aly Deines' independent implementation) → make sure all doctests in this patch are correct (or finish Aly Deines' independent implementation) |
comment:33 Changed 11 years ago by
Description: | modified (diff) |
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Work issues: | make sure all doctests in this patch are correct (or finish Aly Deines' independent implementation) → correct the doctests |
apply trac_9334_nfhilbert.patch
comment:34 Changed 11 years ago by
Status: | needs_work → needs_review |
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Patch applies fine to 4.7.2.alpha2 + #11130; testing now. Meanwhile, just a small point -- in the docstring it says that P must be a prime ideal of self, but it can also be (1) an element of self which generates a prime ideal, or (2) a real or complex place of self. This is well illustrated in the examples, but it should also be stated in the INPUT block.
Any chance you could change the docstring while I am doing the testing?
comment:35 Changed 11 years ago by
Status: | needs_review → needs_work |
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Some doctests disagree with Magma, so I hope they will fail!
comment:36 Changed 11 years ago by
Work issues: | correct the doctests → correct the doctests, change the INPUT in the documentation |
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According to Magma V2.17-9, the following changes make the doctests correct:
replace
sage: K.hilbert_symbol(p,p,p*O) -1
by
sage: K.hilbert_symbol(p,p,p) 1 sage: K.hilbert_symbol(p,3*p,p) -1 sage: K.hilbert_symbol(3,p,p) -1
remove duplicate
sage: K.hilbert_symbol(a,b,P) -1
and for the remaining one, replace -1 by 1
replace -1 by 1 in
sage: K.hilbert_symbol(a, 2, P) -1
Then the "various other examples" contain a lot of uninteresting 1s, of which we can remove a few.
comment:37 Changed 11 years ago by
Testing on 4.7.2.alpha2+#11130 I get these failures:
sage -t -long "devel/sage-main/sage/rings/number_field/number_field.py" ********************************************************************** File "/home/jec/sage-4.7.2.alpha2.11130/devel/sage-main/sage/rings/number_field/number_field.py", line 6737: sage: K.hilbert_symbol(p,p,p*O) Expected: -1 Got: 1 ********************************************************************** File "/home/jec/sage-4.7.2.alpha2.11130/devel/sage-main/sage/rings/number_field/number_field.py", line 6763: sage: K.hilbert_symbol(a,b,P) Expected: -1 Got: 1 ********************************************************************** File "/home/jec/sage-4.7.2.alpha2.11130/devel/sage-main/sage/rings/number_field/number_field.py", line 6770: sage: K.hilbert_symbol(a, b, P) Expected: -1 Got: 1 ********************************************************************** File "/home/jec/sage-4.7.2.alpha2.11130/devel/sage-main/sage/rings/number_field/number_field.py", line 6772: sage: K.hilbert_symbol(a, 2, P) Expected: -1 Got: 1 **********************************************************************
comment:38 follow-up: 40 Changed 11 years ago by
This looks bad:
sage: K.<t> = NumberField(x^3 - x + 1) sage: K.pari_nf().nfhilbert(t,t+1) --------------------------------------------------------------------------- RuntimeError Traceback (most recent call last) /usr/local/src/sage-4.7.2.alpha2/<ipython console> in <module>() /usr/local/src/sage-4.7.2.alpha2/local/lib/python2.6/site-packages/sage/libs/pari/gen.so in sage.libs.pari.gen.gen.nfhilbert (sage/libs/pari/gen.c:29966)() RuntimeError: Segmentation fault
I can't immediately figure out the cause, it is probably a bug somewhere in PARI or the Sage-PARI interface,
comment:40 follow-up: 41 Changed 11 years ago by
Work issues: | correct the doctests, change the INPUT in the documentation → correct the doctests, change the INPUT in the documentation, fix problems with global hilbert symbols, check out seemingly unrelated doctest failures in other files |
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Replying to jdemeyer:
This looks bad:
This is about global Hilbert symbols, while all the doctests are about local ones. Thanks for noticing.
Do you want to close this ticket at the same time as #11130? If not, then I'll wait until I can download a version with #11130 included, and then try to fix the problems here.
comment:41 Changed 11 years ago by
comment:43 follow-up: 44 Changed 11 years ago by
The Segmentation Fault actually shows a major design problem with the Sage->PARI interface (essentially, the t0GEN
system is broken by design).
As for this ticket, as far as I'm concerned you may ignore this. It has essentially nothing to do with this ticket and (hopefully) it can only be reproduced by directly calling PARI (i.e. doing K.pari_nf().nfhilbert(t,t+1)
instead of K.hilbert_symbol(t,t+1)
.)
comment:44 follow-up: 45 Changed 11 years ago by
Replying to jdemeyer:
The Segmentation Fault actually shows a major design problem with the Sage->PARI interface (essentially, the
t0GEN
system is broken by design).
So there should be a new ticket made which explains this major design problem, so that it is on our todo-list at least.
comment:45 Changed 11 years ago by
comment:46 Changed 11 years ago by
Description: | modified (diff) |
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Work issues: | correct the doctests, change the INPUT in the documentation, fix problems with global hilbert symbols, check out seemingly unrelated doctest failures in other files → fix problems/add doctests with global hilbert symbols, check out seemingly unrelated doctest failures in other files |
comment:47 Changed 11 years ago by
Reviewers: | David Loeffler, John Cremona, Marco Streng → David Loeffler, John Cremona, Marco Streng, Jeroen Demeyer |
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comment:48 Changed 11 years ago by
Description: | modified (diff) |
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I presume you didn't mean to apply two copies of the same patch! Is this ready for review again now?
comment:50 follow-ups: 51 52 Changed 11 years ago by
Replying to davidloeffler:
Is this ready for review again now?
There is still a duplicate example, and a lot of examples that return 1 now that they are corrected. I think we should remove some of these, and add some -1's such as the ones in my comment 2 days ago.
Also, what is the output type now? In my patch, I converted it to Integer. Jeroen removed that conversion, but what does cdef long give us?
Finally, in my patch, I had self(a), but Jeroen turned this into a. How carefully does Pari check whether stuff is in the right field?
comment:51 follow-up: 53 Changed 11 years ago by
Replying to mstreng:
Finally, in my patch, I had self(a), but Jeroen turned this into a. How carefully does Pari check whether stuff is in the right field?
Wow, even Sage doesn't check this:
---------------------------------------------------------------------- | Sage Version 4.7.1, Release Date: 2011-08-11 | | Type notebook() for the GUI, and license() for information. | ---------------------------------------------------------------------- sage: K.<a> = NumberField(x^3+x+1) sage: L.<b> = NumberField(x^3+2*x+2) sage: K(b) a
so my self(a)
was pretty useless and we may want to do this:
if not (a in self and b in self): raise ValueError, ...
Changed 11 years ago by
Attachment: | 9334_review_jdemeyer.patch added |
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comment:52 follow-ups: 55 59 Changed 11 years ago by
Authors: | Aly Deines, Marco Streng → Aly Deines, Marco Streng, Jeroen Demeyer |
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Work issues: | fix problems/add doctests with global hilbert symbols, check out seemingly unrelated doctest failures in other files → add/remove some doctests |
Replying to mstreng:
There is still a duplicate example, and a lot of examples that return 1 now that they are corrected. I think we should remove some of these, and add some -1's such as the ones in my comment 2 days ago.
Please do it!
Also, what is the output type now? In my patch, I converted it to Integer. Jeroen removed that conversion, but what does cdef long give us?
It will be Python int
. I see no reason to return a Sage Integer.
Finally, in my patch, I had self(a), but Jeroen turned this into a. How carefully does Pari check whether stuff is in the right field?
I only moved a = self(a)
up in the code.
comment:53 follow-ups: 54 56 Changed 11 years ago by
Replying to mstreng:
---------------------------------------------------------------------- | Sage Version 4.7.1, Release Date: 2011-08-11 | | Type notebook() for the GUI, and license() for information. | ---------------------------------------------------------------------- sage: K.<a> = NumberField(x^3+x+1) sage: L.<b> = NumberField(x^3+2*x+2) sage: K(b) a
I think this is a horrible bug. There is no embedding from L to K! A very generic __call__
method is used, and is definitely not doing the right thing here.
comment:54 Changed 11 years ago by
Replying to cremona:
Replying to mstreng:
---------------------------------------------------------------------- | Sage Version 4.7.1, Release Date: 2011-08-11 | | Type notebook() for the GUI, and license() for information. | ---------------------------------------------------------------------- sage: K.<a> = NumberField(x^3+x+1) sage: L.<b> = NumberField(x^3+2*x+2) sage: K(b) aI think this is a horrible bug. There is no embedding from L to K! A very generic
__call__
method is used, and is definitely not doing the right thing here.
I agree -- that's horrible! It's not so generic actually: the offending code is the method NumberField_absolute._coerce_from_other_number_field
which just converts to a polynomial and back:
f = self.polynomial_ring()(x.polynomial()) return self._element_class(self, f)
This is mathematically meaningless unless either the other field is isomorphic to self, or x is actually in Q. I suggest we raise this on sage-nt, and maybe open a ticket to fix it ASAP.
comment:55 Changed 11 years ago by
Replying to jdemeyer:
It will be Python
int
. I see no reason to return a Sage Integer.
Python ints are fine with me, I was afraid it would be a pari object. I would have liked some uniformity, but that's missing already.
sage: type(legendre_symbol(3,5)) <type 'int'> sage: type(hilbert_symbol(3,5,7)) <type 'sage.rings.integer.Integer'> sage: type(jacobi_symbol(3,5)) <type 'sage.rings.integer.Integer'>
I want all symbols to behave nicely with division by Sage integers, but that's fine with int
sage: int(1)/ZZ(2) 1/2
comment:56 follow-up: 58 Changed 11 years ago by
Replying to cremona:
I think this is a horrible bug. There is no embedding from L to K! A very generic
__call__
method is used, and is definitely not doing the right thing here.
Is there a ticket for this yet?
comment:57 Changed 11 years ago by
Sage-nt thread here: http://groups.google.com/group/sage-nt/browse_thread/thread/9108218411e7f0a6
comment:58 Changed 11 years ago by
comment:59 Changed 11 years ago by
Replying to jdemeyer:
Replying to mstreng:
There is still a duplicate example, and a lot of examples that return 1 now that they are corrected. I think we should remove some of these, and add some -1's such as the ones in my comment 2 days ago.
Please do it!
I could, but I wouldn't be able to test it, and it may need to be rebased afterwards: I failed to install #11130.
comment:60 Changed 11 years ago by
Status: | needs_work → positive_review |
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Positive review for the reviewer patch. I noticed that it included extra -1 examples and removed the duplicate example, and I did not find removing examples worth the trouble.
I also managed to build #11130 and found that all tests pass. I'm assuming Jeroen gives a positive review to what he didn't change, and that his reviewer patch was ready for review, so I'm setting the whole ticket to positive review.
comment:61 Changed 11 years ago by
Milestone: | sage-5.0 → sage-4.7.3 |
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comment:62 Changed 11 years ago by
Work issues: | add/remove some doctests |
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comment:63 Changed 11 years ago by
Milestone: | sage-4.7.3 → sage-pending |
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comment:64 Changed 11 years ago by
Milestone: | sage-pending → sage-4.7.3 |
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comment:66 Changed 11 years ago by
Merged in: | → sage-4.8.alpha1 |
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Milestone: | → sage-4.8 |
Resolution: | → fixed |
Status: | positive_review → closed |
Here all the functions are better placed. I still need to fix the code so that generalized_hilbert_symbol(a,b,P) doesn't assume a.valuation(P) and b.valuation(P) are 0 or 1.