A proof of a conjecture on trace-zero forms and shapes of number fields (Q2221666)

For every finite separable field extension \(L/K\), the bilinear form trace is the map \( T_{L/K}\,:\, L\times L \longrightarrow K \), defined by \(T_{L/K}(x,y)=\displaystyle{\sum_{i=1}^n\sigma_i(xy)}\), where \(\sigma_1,\dots,\sigma_n\) are the \(K\)-isomorphisms of fields from \(L\) to \(\overline{K}\) and \(\overline{K}\) is an algebraic closure of \(K\). This symmetric \(K\)-bilinear form is called the trace form of \(L/K\). When \(L\) is an algebraic number field and \(K= {\mathbb Q}\), the trace form goes back to Hermite and Sylvester. Let \(L\) be a number field. The integral trace form of \(L\) is the restriction of the trace form \(T_{L/{\mathbb Q}}\) to \( {\mathbb Z}_L\times {\mathbb Z}_L \longrightarrow {\mathbb Z} \), where \({\mathbb Z}_L\) is the ring of integers of \(L\). The paper is structured as follows: \begin{enumerate} \item In the first section, the authors introduced the trace zero quadratic module \(({\mathbb Z}_L^0 , T_{L/{\mathbb Q}})\) of a number field \(K\), as the kernel of the trace map; \({\mathbb Z}_L^0= L^0\cap {\mathbb Z}_L\), where \(L^0\) is the kernel of \(T_{L/{\mathbb Q}}\). They also introduced the shape of a number field \(L\), denoted by \(Sh(L)\), as the isometry equivalence class of \(({\mathbb Z}_L^{\perp}, T_{L/{\mathbb Q}})\) up to scalar multiplication, where \({\mathbb Z}_L^{\perp}=\{n\alpha- T_{L/{\mathbb Q}}(\alpha),\, \alpha \in {\mathbb Z}_L\}\) and \(n\) is the degree \([L:{\mathbb Q}]\). \item In the second section, they showed that the integral trace form (resp. the shape) of a number field is a complete invariant for two totally real number fields of fundamental discriminant. \item In the third one, under the separability requirement of extensions, the authors showed that the trace form is also a complete invariant for two extensions to be conjugate. \item In the fourth one, by using Bhargava's parametrization on quartic number fields, the authors went back to prove a conjecture on quartic number fields, which says that: For a totally real number field \(K\) with fundamental discriminant and a number field \(L\), if \(L\) is tamely ramified, then \(L\) and \(K\) are conjugate if and only if their quadratic trace zero modules are isomorphic. More precisely, if \(f:\, {\mathbb Z}_K^0\longrightarrow {\mathbb Z}_L^0\) is isomorphism of rings, which satisfies \(T_{L/{\mathbb Q}}(f(xy))=T_{K/{\mathbb Q}}(xy)\) for every \((x,y)\in K^2\), then \(f\) can be extended to a \({\mathbb Q}\)-isomrphism from \(K\) to \(L\). \end{enumerate}