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Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n 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 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 $x_1$, how can I check that $y$ 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?

Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n 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 $x_1$, how can I check that $y$ 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?

Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n 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 $x_1$, how can I check that 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?

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 $x_1$, 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?

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 $x_1$, 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?