Abelian local \(p\)-class field theory
In the first section an abstract \(p\)-class field theory is developed. Assume that the ground field \(K\) possesses a distinguished Galois extension \(\widetilde {K}/K\) with the Galois group being an abelian pro- \(p\)-group. Then for a finite separable extension \(F/K\) Galois \(p\)- extensions \(L/F\) linearly disjoint with \(\widetilde {F}/F\), \(\widetilde {F}= F\widetilde {K}\) (these extensions are said to be totally ramified) are described by means of a suitable modulation \({\mathcal A}= \{A_ F\): \(F/K\}\). One can introduce a map \(\Upsilon_{L/F}\) acting from the group \(\text{Gal} (L/F)^ \sim\), of continuous \(\mathbb{Z}_ p\)-homomorphisms from \(\text{Gal} (\widetilde {F}/ F)\) to the discrete group \(\text{Gal} (L/F)\), to the quotient group \(U_ F/ N_{L/F} U_ L\), where \(U_ F\) is the kernel of the substantial epimorphism \(v: A_ F\to \mathbb{Z}\). This map is a generalization of the Neukirch map in the classical cases. One of the main conditions on which the theory rests is Hilbert Satz 90 for \(A_ F\). At the same time several other reasonable properties, such as \(\text{Gal} (L/F)\)-stable elements of \(A_ L\) are in one-to-one correspondence with elements of \(A_ F\), are not assumed to hold in general. Thereby, first of all we consider extensions for which a weaker condition: \[ v_{\widetilde {L}} (A_{\widetilde {L}}^{\text{Gal} (\widetilde {L}/\widetilde {F})})= | \widetilde{L} : \widetilde{F}\mid \mathbb{Z} \] holds for a cyclic \(p\)-extension \(\widetilde {L}/ \widetilde {F}\). The class of such extensions (they are called marked) is required to be sufficiently large. For a totally ramified Galois extension \(L/F\), such that any of its intermediate cyclic subextensions \(L_ 1/ F_ 1\) \((F\subset F_ 1\subset L_ 1\subset L)\) is marked, one can define a homomorphism \(\Psi_{L/F}\) (as a generalization of the Hazewinkel homomorphism in one-dimensional case) acting from \(U_ F/ N_{L/F} U_ L\) to \((\text{Gal} (L/F)^{\text{ab}} )^ \sim\); the composition \(\Psi_{L/F} \circ \Upsilon_{L/F}\) is the identity map. Then, assumption on the surjectivity of \(\Upsilon_{L/F}\) for a marked extension of degree \(p\) permits one to show that \(\Upsilon_{L/F}\) and \(\Psi_{L/F}\) are inverse isomorphisms between \((\text{Gal} (L/F)^{\text{ab}} )^ \sim\) and \(U_ F/ N_{L/F} U_ L\). The general case of totally ramified Galois extensions can be handled as well: the definition of \(\Psi_{L/F}\) isn't extendable to arbitrary extensions, but \(\Upsilon_{L/F}\) remains an isomorphism. The next sections contain \(p\)-class field theory for a complete discrete valuation field \(F\) of rank \(n\) with non-algebraically-\(p\)-closed residue field \(k\). Abelian totally ramified (with respect to the discrete valuation of rank \(n\)) \(p\)-extensions are described by means of the subgroup \(VK_ n^{\text{top}} (F)\) in the Milnor topological \(K\)- group \(K_ n^{\text{top}} (F)\), generated by principal units with respect to the discrete valuation. Here for \(n\geq 2\) so-called \(\wp\)- extensions (a tower of subsequent Artin-Schreier extensions) as marked extensions are chosen. The reciprocity map is established: \[ \Psi_ F :VK_ n^{\text{top}} (F)\to \Hom_{\mathbb{Z}_ p} (\text{Gal} (\widetilde {F}/ F), \text{Gal} (F_ p^{\text{ab}}/ \widetilde {F})), \] where \(\widetilde {F}/ F\) is the maximal unramified, with respect to the discrete valuation of rank \(n\), subextension in \(F_ p^{\text{ab}}/ F\), \(F_ p^{\text{ab}}\) is the maximal abelian \(p\)- extension of \(F\). This approach gives, in particular, a new elementary proof of the \(p\)-part of class field theory in the case of higher dimensional local fields with finite residue field. The next sections of the paper contain applications to ramification theory in the case of imperfect residue field (a definition of the Hasse- Herbrand function is exposed); and an existence theorem: description of norm subgroups in \(VK_ n^{\text{top}} (F)\) via the theory of decomposable additive polynomials.
abelian totally ramified \(p\)-extensions
generalization of the Hazewinkel homomorphism
elementary proof of \(p\)-part of class field theory
scientific article; zbMATH DE number 759284
scientific article