LetKbe 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
KasFdecomposes in . More precisely, if is a decomposition ofFinto 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, ifp=2and , we can take

(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

hence . ThenK,i.eit 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 thatfis a monic irreducible polynomial over with a root . so (this also follows directly from (2)). Sincepdoes not divide the index of , it follows from basic algebraic number theory that , where , where is an irreducible decomposition offin .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

Kis also generated by a root ofG.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=2and , then2divides 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 that2is 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 sincecis odd andaanddeven. It follows that , and are orthogonal idempotents of sum 1, thus giving the desired splitting of , hence of .