Abstract
We consider the question of determining whether two binary cubic forms over an arbitrary field
upper K
K
$K$
whose characteristic is not
2
2
$2$
or
3
3
$3$
are equivalent under the actions of either
normal upper G normal upper L left parenthesis 2 comma upper K right parenthesis
G
L
(
2
,
K
)
$\mathrm{GL}(2,K)$
or
normal upper S normal upper L left parenthesis 2 comma upper K right parenthesis
S
L
(
2
,
K
)
$\mathrm{SL}(2,K)$
, deriving two necessary and sufficient criteria for such equivalence in each case. One of these involves an algebraic invariant of binary cubic forms, which we call the Cardano invariant, as it is closely connected to classical formulas; it also appears in the work of Bhargava
et al.
The second criterion is in terms of the base field itself and also gives explicit matrices in
normal upper S normal upper L left parenthesis 2 comma upper K right parenthesis
S
L
(
2
,
K
)
$\mathrm{SL}(2,K)$
or
normal upper G normal upper L left parenthesis 2 comma upper K right parenthesis
G
L
(
2
,
K
)
$\mathrm{GL}(2,K)$
transforming one cubic into the other, if any exist, in terms of the coefficients of bilinear factors of a bicovariant of the two cubics. We also consider automorphisms of a single binary cubic form, show how to use our results to test equivalence of binary cubic forms over an integral domain such as
double struck upper Z
Z
$\mathbb Z$
, and briefly recall some connections between binary cubic forms and the arithmetic of elliptic curves. The methods used are elementary and similar to those used in our work with Fisher concerning equivalences between binary quartic forms.