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- Step 1 will in practice not be necessary since we will always
use this algorithm with
* reduced forms* (see later), which are
automatically irreducible when they belong to **U**.
- Although Step 2 seems to be necessary, this is in fact not the case
since nonprimitive forms will be excluded in the subsequent steps. Indeed,
let
**p** be a prime dividing all the coefficients of **F**. Then clearly
. If **p>3**, Step 4 will return ` false`. Assume now **p=2**
or **p=3**. Then and **F** has at least a double root modulo **p**.
If **F** has only a double and not a triple root, then . On the
other hand if **F** has a triple root, * i.e * if then
with and
hence so and are equal to
zero modulo **p**, hence .
- Even though it seems useful to include the unnecessary Step 2, it
can be shown that on average it
* slows* the algorithm, so in the
final form thanks to the preceding remark we will suppress it.

We end this section by the following proposition.

### Proposition 4.8

Let **K** be a cubic number field, and as above
write where is a fundamental discriminant. Let
be the cubic form associated to **K** by the Davenport-Heilbronn map, and let
its Hessian. Finally, set . Then:
- or and apart from powers of
**3**, **f** and
are squarefree.
- If then .
- If then .
- In particular ,
**f** and have the same prime divisors,
and we can have only equal to , ,
and .

### [Proof]

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