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?