Form the number field, K, which contains a root of the irreducible polynomial q(x)=x3+3x2+3x−2. Name your root a. Verify that q(x) factors, but does not split, over K. With K now as the base field, form an extension of K where the quadratic factor of q(x) has a root. Name this root b, and call this second extension of the tower L.
Use M.<c> = L.absolute\_field() to form the flattened tower that is the absolute number field M. Find the defining polynomial of M with the .polynomial() method. From this polynomial, which must have the generator c as a root, you should be able to use elementary algebra to write the generator as a fairly simple expression.
M should be the splitting field of q(x). To see this, start over, and build from scratch a new number field, P, using the simple expression for c that you just found. Use d as the name of the root used to construct P. Since d is a root of the simple minimal polynomial for c, you should be able to write an expression for d that a pre-calculus student would recognize.
Now factor the original polynomial q(x) (with rational coefficients) over P, to see the polynomial split (as expected). Using this factorization, and your simple expression for d write simplified expressions for the three roots of q(x). See if you can convert between the two versions of the roots ``by hand'', and without using the isomorphisms provided by the .structure() method on M.