Binary Cubic Forms and Cubic Number Fields

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

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