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.Fri, 04 Nov 2022 16:40:49 +0100splitting of primes in extension fieldshttps://ask.sagemath.org/question/64725/splitting-of-primes-in-extension-fields/Call L the maximal real subfield of the cyclotomic field of order 148.
The field L has a quartic subfield — call it K.
I want to see how a prime ideal of K splits in the top field L.
I defined L and K as follows.
sage: C.<a> = CyclotomicField(148)
sage: g = a + a**-1
sage: L.<b> = NumberField(g.minpoly())
sage: subfields = L.subfields()
sage: K = subfields[3][0]
sage: K
Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333
sage: I=K.ideal(37)
sage: F=I.factor()
sage: (Fractional ideal (37, b3))^4
sage: P=list[F]
sage: P
(Fractional ideal (37, b3), 4)
sage: P1= P[0];P1
Fractional ideal (37, b3)
I would like to see how the prime ideal P1 of K factor in the top field L. I tried the following code.
sage: O = L.ring_of_integers()
sage: J = O.P1
But I got the error: No compatible natural embeddings found for Number Field in b with defining polynomial x^36 - 37*x^34 + 629*x^32 - 6512*x^30 + 45880*x^28 - 232841*x^26 + 878787*x^24 - 2510820*x^22 + 5476185*x^20 - 9126975*x^18 + 11560835*x^16 - 10994920*x^14 + 7696444*x^12 - 3848222*x^10 + 1314610*x^8 - 286824*x^6 + 35853*x^4 - 2109*x^2 + 37 and Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333
Could you please help me with this? Thank you.Tue, 01 Nov 2022 17:17:32 +0100https://ask.sagemath.org/question/64725/splitting-of-primes-in-extension-fields/Comment by Rashad for <p>Call L the maximal real subfield of the cyclotomic field of order 148.</p>
<p>The field L has a quartic subfield — call it K.</p>
<p>I want to see how a prime ideal of K splits in the top field L.</p>
<p>I defined L and K as follows.</p>
<pre><code>sage: C.<a> = CyclotomicField(148)
sage: g = a + a**-1
sage: L.<b> = NumberField(g.minpoly())
sage: subfields = L.subfields()
sage: K = subfields[3][0]
sage: K
Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333
sage: I=K.ideal(37)
sage: F=I.factor()
sage: (Fractional ideal (37, b3))^4
sage: P=list[F]
sage: P
(Fractional ideal (37, b3), 4)
sage: P1= P[0];P1
Fractional ideal (37, b3)
</code></pre>
<p>I would like to see how the prime ideal P1 of K factor in the top field L. I tried the following code.</p>
<pre><code>sage: O = L.ring_of_integers()
sage: J = O.P1
</code></pre>
<p>But I got the error: No compatible natural embeddings found for Number Field in b with defining polynomial x^36 - 37<em>x^34 + 629</em>x^32 - 6512<em>x^30 + 45880</em>x^28 - 232841<em>x^26 + 878787</em>x^24 - 2510820<em>x^22 + 5476185</em>x^20 - 9126975<em>x^18 + 11560835</em>x^16 - 10994920<em>x^14 + 7696444</em>x^12 - 3848222<em>x^10 + 1314610</em>x^8 - 286824<em>x^6 + 35853</em>x^4 - 2109<em>x^2 + 37 and Number Field in b3 with defining polynomial x^4 - 37</em>x^2 + 333</p>
<p>Could you please help me with this? Thank you.</p>
https://ask.sagemath.org/question/64725/splitting-of-primes-in-extension-fields/?comment=64764#post-id-64764@slelievre I have edited the question. Could you please look into it? Thank you.Fri, 04 Nov 2022 16:40:49 +0100https://ask.sagemath.org/question/64725/splitting-of-primes-in-extension-fields/?comment=64764#post-id-64764Comment by slelievre for <p>Call L the maximal real subfield of the cyclotomic field of order 148.</p>
<p>The field L has a quartic subfield — call it K.</p>
<p>I want to see how a prime ideal of K splits in the top field L.</p>
<p>I defined L and K as follows.</p>
<pre><code>sage: C.<a> = CyclotomicField(148)
sage: g = a + a**-1
sage: L.<b> = NumberField(g.minpoly())
sage: subfields = L.subfields()
sage: K = subfields[3][0]
sage: K
Number Field in b3 with defining polynomial x^4 - 37*x^2 + 333
sage: I=K.ideal(37)
sage: F=I.factor()
sage: (Fractional ideal (37, b3))^4
sage: P=list[F]
sage: P
(Fractional ideal (37, b3), 4)
sage: P1= P[0];P1
Fractional ideal (37, b3)
</code></pre>
<p>I would like to see how the prime ideal P1 of K factor in the top field L. I tried the following code.</p>
<pre><code>sage: O = L.ring_of_integers()
sage: J = O.P1
</code></pre>
<p>But I got the error: No compatible natural embeddings found for Number Field in b with defining polynomial x^36 - 37<em>x^34 + 629</em>x^32 - 6512<em>x^30 + 45880</em>x^28 - 232841<em>x^26 + 878787</em>x^24 - 2510820<em>x^22 + 5476185</em>x^20 - 9126975<em>x^18 + 11560835</em>x^16 - 10994920<em>x^14 + 7696444</em>x^12 - 3848222<em>x^10 + 1314610</em>x^8 - 286824<em>x^6 + 35853</em>x^4 - 2109<em>x^2 + 37 and Number Field in b3 with defining polynomial x^4 - 37</em>x^2 + 333</p>
<p>Could you please help me with this? Thank you.</p>
https://ask.sagemath.org/question/64725/splitting-of-primes-in-extension-fields/?comment=64744#post-id-64744Ideally, add the definition of the ideal `I`. And check how you tried to define `J`.Wed, 02 Nov 2022 10:53:55 +0100https://ask.sagemath.org/question/64725/splitting-of-primes-in-extension-fields/?comment=64744#post-id-64744