The Heilbronn-Davenport Theorem

We first recall the following results from algebraic number theory.

Proposition 4.1

Let K be a cubic number field of discriminant , and write where is a fundamental discriminant (including 1). Then:

• if and only if p is totally ramified, i.e if and only if .
• implies p=3.
• implies p=3.

[Proof]

We can now state and prove the theorem of Davenport-Heilbronn.

Theorem 4.2

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.

[Proof]

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

Theorem 4.7

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.

• Remarks