(1) has been proved in the preceding section.
(2) 
is a root of 
. It follows from
[3], Chapter 4 Exercise 15, and easily checked anyhow, that
is an order in K, i.e it
is an algebra and a 
-module of finite type, and in particular is a
suborder of the maximal order 
. If 
denotes the 3 roots of
, an easy computation shows that
hence 
.
(3) Assume first that 
. Set
Then f is a monic irreducible polynomial over 
with a root 
.
Now 
and
so 
(this also follows directly from (2)). Since p does
not divide the index of 
, it follows from basic algebraic number
theory that 
, where
, where 
is an irreducible decomposition of f in 
.
But then
for some 
and so
and 
. Finally,
we note that for 
, we have
and the case where
follows.
Assume now that 
. If we can find a matrix
such that
is such that 
, we can apply the preceding
case since K is also generated by a root of G.
One easily checks that if 
but 
we can take
, and if 
, 
but
either 
or 
then there exists
such that 
(any
if 
), and we then take
. This immediately gives the formulas of the
proposition.
Finally, if p=2 and 
, then 2 divides the
coefficient of 
of any form equivalent to 
, and from the definition
of 
this means that 2 divides the index of any 
, in
other words that it is an inessential discriminantal divisor. We then know
that 2 is totally split, and hence 2 still factors as 
modulo 2. To find the factors explicitly, we must split the étale algebra
. All its elements are thus idempotents.
If we set 
, 
and 
considered as elements
of 
(they are in 
, see above), we check that
in 
since c is odd
and a and d even. It follows that 
, 
and 
are
orthogonal idempotents of sum 1, thus giving the desired splitting of
, hence of 
.
