Defining equations and trace forms of cubic cyclic fields (Q1122623)

In this note there are determined the cubic polynomials \(f(X)=X^ 3-aX- b\in {\mathbb{Q}}[X]\) defining cubic cyclic fields in terms of the norm and the trace of some elements of \({\mathbb{Q}}(\rho)\), where \(\rho =-1/2+(3/2)i\). The author studies the quadratic space (E,q), where \(E={\mathbb{Q}}(\alpha)\), \(\alpha\) is a root of f(X) and \(q(x)=Tr_{E/{\mathbb{Q}}}(x^ 2)\) in order to obtain the result.