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# Binary Cubic Forms and Cubic Number Fields

The aim of this section is to generalize to the cubic case the well known correspondence between binary quadratic forms and quadratic number fields. These results are due to Davenport and Heilbronn (see [5] and [6]). Before stating and proving the main theorem, we need a few preliminary results. We let be the set of classes of primitive irreducible binary cubic forms. This makes sense, thanks to Proposition 1.2.

Let K be a cubic number field, and an integral basis of with first element equal to 1. Denote by the discriminant of the number field K. If , denote by , and the conjugates of x in , and let be the discriminant of the (monic) characteristic polynomial of x, so that . Note that if and with and , then

where . Hence, for every , we have .

### Proposition 2.1

Let be an integral basis of a cubic number field K as above. For x and y elements of , set

• For x and y in , we have

• The function is the restriction to of a binary cubic form (again denoted by ) with rational coefficients.
• .
• The form is an integral primitive irreducible cubic form.
• The class of in is independent of the integral basis that we have chosen (hence by abuse of notation we will denote this class by ).
• Let the number field K be defined by a root of the polynomial with p, q, r in , such that there exists an integral basis of the form with t, u, f in and -- this is always possible. Then, when we choose and , we have explicitly:

### [Proof]

Conversely, given an integral primitive irreducible binary cubic form F, we can define a number field associated to F by where is any root of . Since F is irreducible, is an algebraic number of degree exactly equal to 3, so is a cubic field. Choosing another root of F gives an isomorphic (in fact conjugate) field , hence the isomorphism class of is well defined. Finally, if F and G are equivalent under , and are clearly again conjugate. It follows that if we let be the set of isomorphism classes of cubic number fields we have defined maps from to and from to .

### Proposition 2.2

We have , hence is injective and is surjective.

### [Proof]

Let be the image of . It follows from this proposition that and the restriction of to I are inverse discriminant preserving bijections between and I, and this is the Davenport-Heilbronn correspondence that we are looking for. There now remains to determine the image I.

Before doing so, we will show that the form determines the simple invariants of a cubic number field.

### Proposition 2.3

Let K be a cubic field, the associated cubic form, a root of such that (we have seen above that such a exists). Then:
• .
• is an integral basis of .
• A prime decomposes in K as F decomposes in . More precisely, if

is a decomposition of F into irreducible homogeneous factors in then we have

where the are distinct prime ideals of given as follows. Call any lift of in and set . If , then

If but , then

If , , then if or there exists such that (any if ), and then

Finally, if p=2 and , we can take

### [Proof]

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