Before specializing to the cubic case, we consider the general case of
binary forms of degree n. Let K be a field (usually 
, 
or 
).
with
. We will write 
as an abbreviation for
the above notation. In particular 
is the binary quadratic form
, and 
is the cubic form
.
The roots of F in K are the solutions
of 
. If 
for 
and
, the point at infinity 
is a root of order exactly n-m
(if m=n, i.e 
, it is of course not a root), and the other roots
are the roots in K of the polynomial 
of degree m. In particular
if K is algebraically closed, F has always exactly n roots, counted with
multiplicity. Note that the point at infinity is rational over any base field,
algebraically closed or not.
Denote by 
, with 
, the roots
of F in 
. It is easily seen that we can choose representatives in
so that we have
Of course the choice of representative is not unique: for each i we can
change 
into 
as long
as 
. We will always assume that the
representatives of the roots are chosen in this manner.
We define the discriminant of the form F by the following formula:
This makes sense since if we change 
into
with 
,
the product is multiplied by
If 
is a polynomial of degree exactly equal to n ( i.e if
), then one immediately checks that 
.
In degrees up to 3 we have the following formulas.
If F is a form of degree n and 
is
a 
matrix with coefficients in K we define the action of 
on F by 
.
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