Proposition 5.7

Let K be a totally real cubic number field, the unique reduced form associated to K and its Hessian. Then:

• K is cyclic ( i.e ) if and only if .
• if and only if or .
• if and only if or .
• if and only if , or .

Proof

Using Proposition 4.8, this is just a matter of listing reduced quadratic forms and is left to the reader.