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
Letbe an integral basis of a cubic number field K as above. For x and y elements of
, set
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- For x and y in
, we have
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- 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:
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
.
We have, hence
is injective and
is surjective.
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.
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
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