Proposition 4.8

Let K be a cubic number field, and as above write where is a fundamental discriminant. Let be the cubic form associated to K by the Davenport-Heilbronn map, and let its Hessian. Finally, set . Then:

• or and apart from powers of 3, f and are squarefree.
• If then .
• If then .
• In particular , f and have the same prime divisors, and we can have only equal to , , and .

Proof

By Proposition 3.3 (2) we have iff , and by Proposition 2.3 (3) this is true iff p is totally ramified, hence by Proposition 4.1 (1) iff . Hence f and have the same prime divisors. Let p be such a prime divisor. By Davenport-Heilbronn's theorem, hence by Proposition 3.3 (3) we have if , hence if we have and , so up to powers of 3, f is squarefree, and since we have hence up to powers of 3, is also squarefree.

Assume now that p=3 and that p divides f (hence also ). By Proposition 3.3 (3) we have and since we have . Furthermore, if we have hence . Finally, since , using the explicit formulas in terms of the coefficients of the form we see that and , which implies hence . This proves all the assertions of the proposition.