1. Suppose you want to construct a cyclic cubic field. Given a conductor (some number divisible only by primes congruent to 1 mod 3, union {9}), how do you find all fields of that conductor? (Generalises to odd primes.)

 

2. Given an imaginary quadratic field, how do you find all cyclic cubic extensions of it? This is not very difficult, but with a lot more difficulty, I have found the construction for primes 5 and 7.

 

I have found a nice congruence formula to find the primes that split in each of the two cubics with conductor 7*13, for example. And also a way to find the minimal polynomial. This must be known, but isn't found on the internet.

 

When finding extensions of imaginary quadratics, the problem is quite interesting, and there are both the computational and conceptual approaches. The computational approach is to obtain explicit polynomials of degree 3/5/7 with Galois group D6/D10/D14 and subfield say Q(i). Conceptually, these are subfields of torsion point fields of the elliptic curve y^2 = x^3 + x. For Q(sqrt(-3)), the cubics are equivalent to Kummer extensions.