ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 30 May 2016 02:24:48 -0500How make Kummer extensionshttps://ask.sagemath.org/question/33553/how-make-kummer-extensions/ I want to calculate the relative discriminant of field extensions of this kind:
$$\mathbb{Q}(\zeta_5)(\sqrt[5]{a})$$
Where $a \in \mathbb{Q}(\zeta_5)$. So I use SAGE and make this calculations:
K.<b>=CyclotomicField(5); //my field base
alpha=1+3*b^2; //an element of my field base
f=(1+3*b^2).minpoly(); //its minimal polynomial
f.is_irreducible() //is it irreducible?
R.<a>=K.extension(f) //the field extension of my field base
R.relative_discriminant() //the calculation of the relative discriminant
But when I execute it, appears this error
defining polynomial (x^4 - x^3 + 6*x^2 + 14*x + 61) must be irreducible
But it is irreducible, what am I doing wrong? Or how can I solve this?Thu, 26 May 2016 08:10:12 -0500https://ask.sagemath.org/question/33553/how-make-kummer-extensions/Answer by A.Alharbi for <p>I want to calculate the relative discriminant of field extensions of this kind:
$$\mathbb{Q}(\zeta_5)(\sqrt[5]{a})$$
Where $a \in \mathbb{Q}(\zeta_5)$. So I use SAGE and make this calculations:</p>
<pre><code>K.<b>=CyclotomicField(5); //my field base
alpha=1+3*b^2; //an element of my field base
f=(1+3*b^2).minpoly(); //its minimal polynomial
f.is_irreducible() //is it irreducible?
R.<a>=K.extension(f) //the field extension of my field base
R.relative_discriminant() //the calculation of the relative discriminant
</code></pre>
<p>But when I execute it, appears this error</p>
<pre><code>defining polynomial (x^4 - x^3 + 6*x^2 + 14*x + 61) must be irreducible
</code></pre>
<p>But it is irreducible, what am I doing wrong? Or how can I solve this?</p>
https://ask.sagemath.org/question/33553/how-make-kummer-extensions/?answer=33557#post-id-33557Note: I have NOT studied cyclotomic fields.
<strike> I believe the problem lies in `minpoly()` function. It has been [reported](http://ask.sagemath.org/question/23915/minimal-polynomial-isnt-minimal/) that `minpoly` sometimes doesn not return an irreducible polynomial.</strike>
Also, if you check where `f` belongs, you will see that it is not in $\mathbb{Q}(\xi)_5[x]$ !.
sage: K.<b>=CyclotomicField(5);
sage: alpha=1+3*b^2;
sage: f=(1+3*b^2).minpoly()
sage: f.parent()
Univariate Polynomial Ring in x over Rational Field
sage: factor(f)
x^4 - x^3 + 6*x^2 + 14*x + 61
sage: G.<x> = K[] #Define a polynomial ring over K
sage: f = G(f) #Coerce f into G
sage: f.parent()
Univariate Polynomial Ring in x over Cyclotomic Field of order 5 and degree 4
sage: factor(f)
(x - 3*b - 1) * (x - 3*b^2 - 1) * (x - 3*b^3 - 1) * (x + 3*b^3 + 3*b^2 + 3*b + 2)
sage: f.is_irreducible()
False
______
**Update**:
Could you please elaborate which element would you like to append it to $\mathbb{Q}(\xi)_5$?
It seems to me that $\alpha$ is already in $\mathbb{Q}(\xi)_5$ but $\sqrt[5]{\alpha}$ is not. If you use fractional expontiation or the function `nth_root` in `minpoly`, it will gives you an error. I am afraid that you have to hardcode the equation $x^5 - \alpha$
sage: alpha in K
True
sage: alpha^(1/5) in K
False
sage: L.<a> = K.extension(x^5 - alpha)
sage: L
Number Field in a with defining polynomial x^5 - 3*b^2 - 1 over its base field
_____
**Second update**
> If you use fractional exponentiation or the function `nth_root` in `minpoly`, it will gives you an error.
It seems an issue need to be solved.
> by the way, what f = G(f) means?
$f$ was living in $\mathbb{Q}$,i.e. arithmetic on $f$ would be carry over $\mathbb{Q}$, writing `f = G(f)` tells sage $f$ is an element of $G$
Example:
sage: a = 7
sage: a^2
49
But,
sage: R = IntegerModRing(13)
sage: a = R(a)
sage: a
7
sage: a^2
10
sage: a + 6
0
For more information [see](http://doc.sagemath.org/html/en/tutorial/tour_coercion.html)
Thu, 26 May 2016 14:31:19 -0500https://ask.sagemath.org/question/33553/how-make-kummer-extensions/?answer=33557#post-id-33557Comment by nbruin for <p>Note: I have NOT studied cyclotomic fields.</p>
<p><strike> I believe the problem lies in <code>minpoly()</code> function. It has been <a href="http://ask.sagemath.org/question/23915/minimal-polynomial-isnt-minimal/">reported</a> that <code>minpoly</code> sometimes doesn not return an irreducible polynomial.</strike> </p>
<p>Also, if you check where <code>f</code> belongs, you will see that it is not in $\mathbb{Q}(\xi)_5[x]$ !. </p>
<pre><code>sage: K.<b>=CyclotomicField(5);
sage: alpha=1+3*b^2;
sage: f=(1+3*b^2).minpoly()
sage: f.parent()
Univariate Polynomial Ring in x over Rational Field
sage: factor(f)
x^4 - x^3 + 6*x^2 + 14*x + 61
sage: G.<x> = K[] #Define a polynomial ring over K
sage: f = G(f) #Coerce f into G
sage: f.parent()
Univariate Polynomial Ring in x over Cyclotomic Field of order 5 and degree 4
sage: factor(f)
(x - 3*b - 1) * (x - 3*b^2 - 1) * (x - 3*b^3 - 1) * (x + 3*b^3 + 3*b^2 + 3*b + 2)
sage: f.is_irreducible()
False
</code></pre>
<hr/>
<p><strong>Update</strong>:
Could you please elaborate which element would you like to append it to $\mathbb{Q}(\xi)_5$?
It seems to me that $\alpha$ is already in $\mathbb{Q}(\xi)_5$ but $\sqrt[5]{\alpha}$ is not. If you use fractional expontiation or the function <code>nth_root</code> in <code>minpoly</code>, it will gives you an error. I am afraid that you have to hardcode the equation $x^5 - \alpha$</p>
<pre><code>sage: alpha in K
True
sage: alpha^(1/5) in K
False
sage: L.<a> = K.extension(x^5 - alpha)
sage: L
Number Field in a with defining polynomial x^5 - 3*b^2 - 1 over its base field
</code></pre>
<hr/>
<p><strong>Second update</strong></p>
<blockquote>
<p>If you use fractional exponentiation or the function <code>nth_root</code> in <code>minpoly</code>, it will gives you an error.</p>
</blockquote>
<p>It seems an issue need to be solved.</p>
<blockquote>
<p>by the way, what f = G(f) means?</p>
</blockquote>
<p>$f$ was living in $\mathbb{Q}$,i.e. arithmetic on $f$ would be carry over $\mathbb{Q}$, writing <code>f = G(f)</code> tells sage $f$ is an element of $G$</p>
<p>Example:</p>
<pre><code>sage: a = 7
sage: a^2
49
</code></pre>
<p>But,</p>
<pre><code>sage: R = IntegerModRing(13)
sage: a = R(a)
sage: a
7
sage: a^2
10
sage: a + 6
0
</code></pre>
<p>For more information <a href="http://doc.sagemath.org/html/en/tutorial/tour_coercion.html">see</a></p>
https://ask.sagemath.org/question/33553/how-make-kummer-extensions/?comment=33570#post-id-33570minpoly behaves as it should in this case. minpoly(alpha) is irreducible over Q, and (alpha being a root of it), it does have a root over K.
In your update you suggest the proper approach: adjoin the *fifth root* of alpha by constructing a polynomial that has the appropriate root.Sat, 28 May 2016 13:23:40 -0500https://ask.sagemath.org/question/33553/how-make-kummer-extensions/?comment=33570#post-id-33570Answer by belvedere for <p>I want to calculate the relative discriminant of field extensions of this kind:
$$\mathbb{Q}(\zeta_5)(\sqrt[5]{a})$$
Where $a \in \mathbb{Q}(\zeta_5)$. So I use SAGE and make this calculations:</p>
<pre><code>K.<b>=CyclotomicField(5); //my field base
alpha=1+3*b^2; //an element of my field base
f=(1+3*b^2).minpoly(); //its minimal polynomial
f.is_irreducible() //is it irreducible?
R.<a>=K.extension(f) //the field extension of my field base
R.relative_discriminant() //the calculation of the relative discriminant
</code></pre>
<p>But when I execute it, appears this error</p>
<pre><code>defining polynomial (x^4 - x^3 + 6*x^2 + 14*x + 61) must be irreducible
</code></pre>
<p>But it is irreducible, what am I doing wrong? Or how can I solve this?</p>
https://ask.sagemath.org/question/33553/how-make-kummer-extensions/?answer=33580#post-id-33580Ok, what is happening is that minoply returns de minimal polynomial over $\mathbb{Q}$ and not over a specific field. This answered me.
by the way, what
> f = G(f)
means?
Mon, 30 May 2016 02:24:48 -0500https://ask.sagemath.org/question/33553/how-make-kummer-extensions/?answer=33580#post-id-33580