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, 19 Feb 2024 12:14:14 +0100Cyclotomic fields - displaying powers of zetahttps://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/Is there any way how to avoid expressing $\zeta^{n-1}$ in terms of lower powers of $\zeta$ when working with cyclotomic fields?
K.<zeta> = CyclotomicField(3)
zeta^5
-zeta - 1
I would like to show $\zeta^2$ in this case.
I know that I can do this instead:
K.<zeta> = QQ[]
zeta^5 % (zeta^3 - 1)
zeta^2
But the problem is that eventually, I want to work with a polynomial ring over K and perform a factorization there, and that is not possible in `QQ[].<x,y>`.Sun, 18 Feb 2024 20:39:43 +0100https://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/Comment by Max Alekseyev for <p>Is there any way how to avoid expressing $\zeta^{n-1}$ in terms of lower powers of $\zeta$ when working with cyclotomic fields?</p>
<pre><code>K.<zeta> = CyclotomicField(3)
zeta^5
-zeta - 1
</code></pre>
<p>I would like to show $\zeta^2$ in this case.</p>
<p>I know that I can do this instead:</p>
<pre><code>K.<zeta> = QQ[]
zeta^5 % (zeta^3 - 1)
zeta^2
</code></pre>
<p>But the problem is that eventually, I want to work with a polynomial ring over K and perform a factorization there, and that is not possible in <code>QQ[].<x,y></code>.</p>
https://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/?comment=76084#post-id-76084Then work modulo `zeta^3 - 1` while you don't do factoring; and when you do, embed the argument into the cyclotomic field first.Mon, 19 Feb 2024 12:14:14 +0100https://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/?comment=76084#post-id-76084Comment by kejv2 for <p>Is there any way how to avoid expressing $\zeta^{n-1}$ in terms of lower powers of $\zeta$ when working with cyclotomic fields?</p>
<pre><code>K.<zeta> = CyclotomicField(3)
zeta^5
-zeta - 1
</code></pre>
<p>I would like to show $\zeta^2$ in this case.</p>
<p>I know that I can do this instead:</p>
<pre><code>K.<zeta> = QQ[]
zeta^5 % (zeta^3 - 1)
zeta^2
</code></pre>
<p>But the problem is that eventually, I want to work with a polynomial ring over K and perform a factorization there, and that is not possible in <code>QQ[].<x,y></code>.</p>
https://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/?comment=76079#post-id-76079Basically I want a structure where I have both `zeta^2` (or at least for display) and ability to factor in the polyring.Mon, 19 Feb 2024 08:28:26 +0100https://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/?comment=76079#post-id-76079Comment by Max Alekseyev for <p>Is there any way how to avoid expressing $\zeta^{n-1}$ in terms of lower powers of $\zeta$ when working with cyclotomic fields?</p>
<pre><code>K.<zeta> = CyclotomicField(3)
zeta^5
-zeta - 1
</code></pre>
<p>I would like to show $\zeta^2$ in this case.</p>
<p>I know that I can do this instead:</p>
<pre><code>K.<zeta> = QQ[]
zeta^5 % (zeta^3 - 1)
zeta^2
</code></pre>
<p>But the problem is that eventually, I want to work with a polynomial ring over K and perform a factorization there, and that is not possible in <code>QQ[].<x,y></code>.</p>
https://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/?comment=76073#post-id-76073`zeta^5 % (zeta^3 - 1)` is not the same as computing `zeta^5` in `CyclotomicField(3)` as the latter is done modulo cyclotomic polynomial `zeta^2 + zeta + 1`. So it's unclear what you want.Sun, 18 Feb 2024 22:55:12 +0100https://ask.sagemath.org/question/76070/cyclotomic-fields-displaying-powers-of-zeta/?comment=76073#post-id-76073