Let K be a cubic number field of discriminant , and write where is a fundamental discriminant (including 1). Then:
We can now state and prove the theorem of Davenport-Heilbronn.
We have . In other words, the maps and are discriminant preserving inverse bijections between isomorphism classes of cubic fields and binary cubic forms belonging to U.
As a consequence of this theorem and of the results of the previous section, we now have an effficient algorithm to test whether a cubic form corresponds to the image of a cubic field by the Davenport-Heilbronn map as follows
Let be a cubic form. This algorithm outputs true or false according to whether F corresponds to the image of a cubic field by the Davenport-Heilbronn map.
- If F is not irreducible, return false.
- If F is not primitive, return false.
- Compute the Hessian by the formulas given. Set , (so ). Using Proposition 3.3, check that and , otherwise return false.
- If for some p>3, return false.
- Set . Remove all powers of 2 and 3 from t (in fact at most and ), and let again t be the result. If return false.
- If t is squarefree return true, otherwise return false.