Integral trace forms associated to cubic extensions

Details Integral trace forms associated to cubic extensions Integral trace forms associated to cubic extensions

2011-04-24

arxiv.org/abs/1104.4598v1

Algebra Number Theory 4 (2010), no. 6, 681-699

Mathematics

Number Theory

This is a slightly different version from the published one(There are only format differences, and correction of two typos but

Scientific paper

Given a nonzero integer $d$, we know by Hermite's Theorem that there exist only finitely many cubic number fields of discriminant $d$. However, it can happen that two non-isomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form $q_K:\text{tr}_{K/\mathbb{Q}}(x^2)|_{O^{0}_{K}}$ as such a refinement. For a cubic field of fundamental discriminant $d$ we show the existence of an element $T_K$ in Bhargava's class group $\Cl(\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}; -3d)$ such that $q_K$ is completely determined by $T_K$. By using one of Bhargava's composition laws, we show that $q_K$ is a complete invariant whenever $K$ is totally real and of fundamental discriminant

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