ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 09 Oct 2023 06:48:13 +0200How can I obtain representatives of a quotient ring?https://ask.sagemath.org/question/73816/how-can-i-obtain-representatives-of-a-quotient-ring/ I want to compute quotient of integer ring of $\mathbb{Q}(\omega) (\omega^3=1)$ by a prime ideal $(-4-3\omega)$. Especially I want to compute representatives.
N=3
x=polygen(ZZ,'x')
K.<a>=CyclotomicField(N)
O = K.ring_of_integers()
p=-4-3*a
R.<b,c>=QuotientRing(O, K.ideal(p))
What should I do next?
More, I want to compute its cardinality (=13), But
R.cardinality()
made error.Sun, 08 Oct 2023 10:52:01 +0200https://ask.sagemath.org/question/73816/how-can-i-obtain-representatives-of-a-quotient-ring/Answer by rburing for <p>I want to compute quotient of integer ring of $\mathbb{Q}(\omega) (\omega^3=1)$ by a prime ideal $(-4-3\omega)$. Especially I want to compute representatives.</p>
<pre><code>N=3
x=polygen(ZZ,'x')
K.<a>=CyclotomicField(N)
O = K.ring_of_integers()
p=-4-3*a
R.<b,c>=QuotientRing(O, K.ideal(p))
</code></pre>
<p>What should I do next?</p>
<p>More, I want to compute its cardinality (=13), But</p>
<pre><code>R.cardinality()
</code></pre>
<p>made error.</p>
https://ask.sagemath.org/question/73816/how-can-i-obtain-representatives-of-a-quotient-ring/?answer=73820#post-id-73820Go this way instead:
sage: P = K.ideal(p)
sage: R = P.residue_field()
sage: R.cardinality()
13
sage: [z.lift() for z in R]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
Why representatives are integers:
sage: R(a)
3
sage: (-a)*p
a - 3
sage: (-a)*P in P
True
So in the quotient $a = 3$.
In general $a$ is mapped to one of `K.defining_polynomial().roots(R, multiplicities=False)`.Sun, 08 Oct 2023 12:08:29 +0200https://ask.sagemath.org/question/73816/how-can-i-obtain-representatives-of-a-quotient-ring/?answer=73820#post-id-73820Comment by Ys1123 for <p>Go this way instead:</p>
<pre><code>sage: P = K.ideal(p)
sage: R = P.residue_field()
sage: R.cardinality()
13
sage: [z.lift() for z in R]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
</code></pre>
<p>Why representatives are integers:</p>
<pre><code>sage: R(a)
3
sage: (-a)*p
a - 3
sage: (-a)*P in P
True
</code></pre>
<p>So in the quotient $a = 3$.</p>
<p>In general $a$ is mapped to one of <code>K.defining_polynomial().roots(R, multiplicities=False)</code>.</p>
https://ask.sagemath.org/question/73816/how-can-i-obtain-representatives-of-a-quotient-ring/?comment=73837#post-id-73837Thank you. What does R(a) mean?
By the way this way seems very ad hoc. Isnâ€™t there any ways to compute representatives of quotient of general (polynomial) ring?Mon, 09 Oct 2023 06:48:13 +0200https://ask.sagemath.org/question/73816/how-can-i-obtain-representatives-of-a-quotient-ring/?comment=73837#post-id-73837