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, 15 Apr 2022 17:07:40 +0200How can i know if a field is monigenic?https://ask.sagemath.org/question/61974/how-can-i-know-if-a-field-is-monigenic/I know the next field is monogenic but SageMath doesn't give me a power basis.
sage: c = sqrt(-(5+2*sqrt(5)))
sage: L.<c> = QQ[c]
sage: OL = L.maximal_order()
sage: B = OL.basis()
sage: L
Number Field in a with defining polynomial x^4 + 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?*I
sage: OL
Maximal Order in Number Field in a with defining polynomial x^4 + 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?*I
sage: B
[3/8*a^3 + 3/8*a^2 + 1/8*a + 1/8, 3/4*a^3 + 1/4*a, 1/2*a^3 + 1/2*a^2, a^3]
sage: L.integral_basis()
[3/8*a^3 + 3/8*a^2 + 1/8*a + 1/8, 3/4*a^3 + 1/4*a, 1/2*a^3 + 1/2*a^2, a^3]Fri, 15 Apr 2022 02:16:16 +0200https://ask.sagemath.org/question/61974/how-can-i-know-if-a-field-is-monigenic/Comment by rburing for <p>I know the next field is monogenic but SageMath doesn't give me a power basis.</p>
<pre><code>sage: c = sqrt(-(5+2*sqrt(5)))
sage: L.<c> = QQ[c]
sage: OL = L.maximal_order()
sage: B = OL.basis()
sage: L
Number Field in a with defining polynomial x^4 + 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?*I
sage: OL
Maximal Order in Number Field in a with defining polynomial x^4 + 10*x^2 + 5 with a = 0.?e-18 + 3.077683537175254?*I
sage: B
[3/8*a^3 + 3/8*a^2 + 1/8*a + 1/8, 3/4*a^3 + 1/4*a, 1/2*a^3 + 1/2*a^2, a^3]
sage: L.integral_basis()
[3/8*a^3 + 3/8*a^2 + 1/8*a + 1/8, 3/4*a^3 + 1/4*a, 1/2*a^3 + 1/2*a^2, a^3]
</code></pre>
https://ask.sagemath.org/question/61974/how-can-i-know-if-a-field-is-monigenic/?comment=61993#post-id-61993The method from section 5 of [Computing power integral bases
in algebraic number fields](https://www.degruyter.com/document/doi/10.1515/9783110809794.243/pdf) yields that the powers of `(c + 1)/2` form a basis of `OL`. I don't know if there is a general method for fields of arbitrary degree, so are you interested specifically in quartic fields?Fri, 15 Apr 2022 17:07:40 +0200https://ask.sagemath.org/question/61974/how-can-i-know-if-a-field-is-monigenic/?comment=61993#post-id-61993