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On algebraic number fields in Sage

Assume I have an irreducible monic polynomial f over 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 x1,...,xn, and I define y to be a linear combination of x1, 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=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 xi whether O=Z[y]?

Question 3: How can I check via Sage whether O is norm-Euclidean?

On algebraic number fields in Sage

Assume I have an irreducible monic polynomial f over 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 x1,...,xn, and I define y to be a linear combination of x1, 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=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 xi whether O=Z[y]?

Question 3: How can I check via Sage whether O is norm-Euclidean?

On algebraic number fields in Sage

Assume I have an irreducible monic polynomial f over 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 x1,...,xn, and I define y to be a linear combination of x1, 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=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 xi whether O=Z[y]?

Question 3: How can I check via Sage whether O is norm-Euclidean?

On algebraic number fields in Sage

Assume I have an irreducible monic polynomial f over Z of degree n (take for example f=x4+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 x1,...,xn, and I define y to be a linear combination of x1, 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=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 xi whether O=Z[y]?

Question 3: How can I check via Sage whether O is norm-Euclidean?

On algebraic number fields in Sage

Assume I have an irreducible monic polynomial f over Z of degree n (take for example f=x4+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 x1,...,xn, and I define y to be a linear combination of x1, the xi, 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=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 xi whether O=Z[y]?

Question 3: How can I check via Sage whether O is norm-Euclidean?