Spectral norms on valued fields (Q5949444)

Let \((K,|.|)\) be a perfect valued field, \(\bar{K}\) an algebraic closure of \(K\) and \(|.|\) the extension to \(\bar{K}\). Let \(G\) be the group of all \(K\)-automorphisms of \(\bar{K}\) and \(||x||:= \sup \{|\sigma x|\mid \sigma \in G\}\), the \(G\)-spectral norm on \(\bar K\). Let \(\tilde{\bar{K}}\) be the completion of \(\bar K\) with respect to \(||.||\). For an algebraic extension \(K\subset L\subset \bar K\), let \(\tilde L\) be the closure of \(L\) in \(\tilde{\bar{K}}\). After some preliminaries, the authors show in section 2 that \(\tilde{L} \cap \bar{K} = L\). For any \(x\in \tilde{\bar{K}}\) there is an invariant \(\omega(x)\in {\mathbb R}\), \(\omega (x) \geq 0\) associated to \(x\). It is proved that \(\omega(x)=0\) if and only if \(x\in \tilde{K}\). With some additional hypothesis the authors prove that \(\tilde{L}\) is algebraic over \(\tilde{K}\). In section 3 the notion of minimal generating field is shown for an element \(x \in \tilde{\bar{K}}\) and is shown the existence of a minimal generating field for any \(x\in \tilde{\bar{K}}\). Next, the authors give a characterization of the elements \(x\in \tilde{\bar{K}}\) which admit finite generating fields and show that not all \(x\in \tilde{\bar{K}}\) have finite generating field. Furthermore, they give a complete description of the valued fields \((K,|.|)\) such that any element \(x\in \tilde{\bar{K}}\) has a finite generating field. In the last section, it is proved that if \([\tilde{K}L : \tilde{K}]< \infty\), then \(\tilde{L}\) is a zero-dimensional regular ring and as an application it is proved that \(\tilde{\bar{{\mathbb Q}}}\) is an algebraically closed zero-dimensional regular ring.