Real cubic fields

Definition 5.1 Let be a quadratic form with real coefficients. We will say that H is reduced if and R>0 (to exclude the trivial case of the zero form).

Note that this is stronger than the usual definition, which would be with when one of the inequalities is an equality. The reason for the modification to the usual definition is that we must work with forms modulo and not only .

That the above definition works follows from the following lemma.

Lemma 5.2

Let and be two reduced definite integral binary quadratic forms such that there exists with . Then . Furthermore

these cases being inclusive.

[Proof]

We can now give the definition of a reduced cubic form in the case of positive discriminant.

Definition 5.3 Let be an integral binary cubic form of positive discriminant. We will say that F is reduced if its Hessian is reduced in the above sense, and if in addition:

• a>0.
• If Q=0, then b>0.
• If P=Q, then |b|<|3a-b|.
• If P=R, then and |b|<|c| when |d|=a.

With this definition, we have:

Proposition 5.4

• Two equivalent reduced real cubic forms are equal.
• A reduced real cubic form belonging to U is irreducible.
• Any irreducible real cubic form is equivalent to a unique reduced form.

[Proof]

To be able to produce all reduced binary cubic forms of discriminant bounded by X, we must be able to give bounds on the coefficients of such a form. Such a result is as follows.

Proposition 5.5

Let be a reduced form such that . We have the following inequalities:

[Proof]

Using only the inequalities of Proposition 5.5, it is not difficult to show that the number of quadruplets which will have to be checked is linear in X. However, in the actual implementation, we will first loop on a, then on b, then on c and finally on d satisfying the inequalities coming from and from . The total number of triplets satisfying the above inequalities is . For some of these triplets the loop on d will be empty, and for the others we will have to examine essentially the number of reduced real binary cubic forms of discriminant up to X (essentially because the special cases have to be considered, but they add a negligible number of forms). Hence the total number of cases to be examined will be . Now recall the following theorem.

Theorem 5.6

Let (resp. ) be the number of equivalent real cubic forms (resp. of isomorphism classes of real cubic fields) of discriminant less than or equal to X. Then as , we have:

for any .

• Remark