ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 08 Mar 2015 14:00:23 -0500Ideal moduli and residue symbolshttp://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/ Hi, everyone;
I'm fairly new to sage, but I feel like I have some heavy lifting to do. I'm attempting to do a couple of things:
1) I need to see if two complex numbers are equivalent mod an ideal, eg pi == 1 mod 2+2i. It might be a dumb question but my searching thus far has come up short
2) I need to compute residue symbols, and I'm using the Number Field residue symbol method, but I'm having trouble. I have the following:
C = ComplexField()
I = C.0
r = C.ideal(b).residue_symbol(D,4)
with a and b complex numbers. Help!Thu, 05 Mar 2015 19:37:25 -0600http://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/Comment by jwiebe for <p>Hi, everyone;</p>
<p>I'm fairly new to sage, but I feel like I have some heavy lifting to do. I'm attempting to do a couple of things:</p>
<p>1) I need to see if two complex numbers are equivalent mod an ideal, eg pi == 1 mod 2+2i. It might be a dumb question but my searching thus far has come up short</p>
<p>2) I need to compute residue symbols, and I'm using the Number Field residue symbol method, but I'm having trouble. I have the following:
C = ComplexField()
I = C.0
r = C.ideal(b).residue_symbol(D,4)
with a and b complex numbers. Help!</p>
http://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/?comment=26055#post-id-26055Okay, after reading a bit more I can clarify. The ambient space is the ring of integers of Q(zeta_m) - mth root of unity. So these ideals are in the ring.Fri, 06 Mar 2015 15:00:57 -0600http://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/?comment=26055#post-id-26055Comment by tmonteil for <p>Hi, everyone;</p>
<p>I'm fairly new to sage, but I feel like I have some heavy lifting to do. I'm attempting to do a couple of things:</p>
<p>1) I need to see if two complex numbers are equivalent mod an ideal, eg pi == 1 mod 2+2i. It might be a dumb question but my searching thus far has come up short</p>
<p>2) I need to compute residue symbols, and I'm using the Number Field residue symbol method, but I'm having trouble. I have the following:
C = ComplexField()
I = C.0
r = C.ideal(b).residue_symbol(D,4)
with a and b complex numbers. Help!</p>
http://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/?comment=26050#post-id-26050Could you please define what you mean by "ideal". The complex plane is a field, so every ideal in the usual sense is trivial here.Fri, 06 Mar 2015 11:36:00 -0600http://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/?comment=26050#post-id-26050Answer by vdelecroix for <p>Hi, everyone;</p>
<p>I'm fairly new to sage, but I feel like I have some heavy lifting to do. I'm attempting to do a couple of things:</p>
<p>1) I need to see if two complex numbers are equivalent mod an ideal, eg pi == 1 mod 2+2i. It might be a dumb question but my searching thus far has come up short</p>
<p>2) I need to compute residue symbols, and I'm using the Number Field residue symbol method, but I'm having trouble. I have the following:
C = ComplexField()
I = C.0
r = C.ideal(b).residue_symbol(D,4)
with a and b complex numbers. Help!</p>
http://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/?answer=26077#post-id-26077It is fairly straightforward once you know Sage standards
sage: K = CyclotomicField(9)
sage: O = K.ring_of_integers()
sage: zeta9 = O.gen(1)
At this point you have three objects defined in the console/notebook: the cyclotomic field **K**, its ring of integers **O** and the generator **zeta9**. Now you can defined ideals and quotients as follows.
sage: I = O.ideal(3*zeta9^2 + 2*zeta9^3 + 5)
sage: R = O.quotient(I, 'a')
note: I am not sure why, but the arguemnt 'a' is mandatory in the method **quotient**.
To check equality modulo **I** just do it in the quotient
sage: R(3*zeta9^2 + 7) == R(2 - 2*zeta9^3)
True
sage: R(zeta9) == R(2)
False
VincentSun, 08 Mar 2015 14:00:23 -0500http://ask.sagemath.org/question/26042/ideal-moduli-and-residue-symbols/?answer=26077#post-id-26077