Contents
** Next:** Remarks
**Up:** Binary Cubic Forms and
** Previous:** Algorithmic Characterization of

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**.

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**.

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`.

### [Proof]

Contents
** Next:** Remarks
**Up:** Binary Cubic Forms and
** Previous:** Algorithmic Characterization of