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Before specializing to the cubic case, we consider the general case of
binary forms of degree **n**. Let **K** be a field (usually , or ).
A * binary form* of degree **n** with coefficients in **K** is a homogeneous
polynomial in two variables of degree **n** with coefficients in **K**, * i.e * an
expression of the form

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 .

### Proposition 1.1

Let . Then

We now restrict to the case of * integral* binary forms, * i.e * binary
forms whose coefficients are all in . As a corollary of the proposition,
we see that the action of preserves the discriminant of **F**,
since is even.

We will say that the form **F** is * irreducible* if is
irreducible as a polynomial in . Equivalently, **F** is irreducible if
and the polynomial is irreducible in (or ).

We will say that an integral form **F** is * primitive* if the GCD of all
its coefficients is equal to 1.

### Proposition 1.2

Let **F** be an integral form and .
Then is irreducible if and only if **F** is, and
is primitive if and only **F** is.

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