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, 08 Mar 2021 19:24:02 +0100On algebraic number fields in Sagehttps://ask.sagemath.org/question/56061/on-algebraic-number-fields-in-sage/Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n (take for example $f=x^4+x+1$) and let $K$ be the algebraic number field associated to it and $O$ the ring of integers in $K$. Assume that $K$ has solveable Galois group so that in principle we (can) know that roots of the polynomial.
>Question 1: How can I obtain all the roots of the polynomial $f$ via Sage? If they are called $x_1,...,x_n$, and I define $y$ to be a linear combination of the $x_i$, how can I check whehter $y$ is a primitive element for $K$ via Sage?
>Question 2: How can I check whether $O$ is monogenic, that is $O=\mathbb{Z}[y]$ for some element $y$? As in question 1, how can I check for a specific $y$ that is a linear combination of the $x_i$ whether $O=\mathbb{Z}[y]$?
>Question 3: How can I check via Sage whether $O$ is norm-Euclidean?Mon, 08 Mar 2021 19:24:02 +0100https://ask.sagemath.org/question/56061/on-algebraic-number-fields-in-sage/