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Necessary conditions for local class field theory
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Grundlehren Der Mathematischen Wissenschaften in Einzeldarstellungen Mit Besonderer Berücksichtigung Der Anwendungsgebiete, 1986
The abstract class field theory that we have developed in the last chapter is now going to be applied to the case of a local field, i.e., to a field which is complete with respect to a discrete valuation, and which has a finite residue class field. By chap. II, (5.2), these are precisely the finite extensions K of the fields ℚ p or F p ((t)).
Jürgen Neukirch
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The abstract class field theory that we have developed in the last chapter is now going to be applied to the case of a local field, i.e., to a field which is complete with respect to a discrete valuation, and which has a finite residue class field. By chap. II, (5.2), these are precisely the finite extensions K of the fields ℚ p or F p ((t)).
Jürgen Neukirch
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Graduate Texts in Mathematics, 1979
Standard local class field theory is concerned with complete fields K whose residue field is finite.
Jean-Pierre Serre
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Standard local class field theory is concerned with complete fields K whose residue field is finite.
Jean-Pierre Serre
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Local Class Field Theory: Lubin–Tate Theory
Universitext, 2020This chapter covers local class field theory via Lubin–Tate construction, giving a third proof of the local existence theorem.
David Harari
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Part II Local Class Field Theory
2013Local and global class field theory, as well as a series of further theories for which the name class field theory is similarly justified, have the following principle in common. All of these theories involve a canonical bijective correspondence between the abelian extensions of a field K and certain subgroups of a corresponding module AK associated ...
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The Annals of Mathematics, 1950
Local class field theory is treated by means of cohomology theory. Let \(L/K\) be a Galois extension with Galois group \(\mathfrak L\). Let \(\mathfrak H\) be an invariant subgroup of \(\mathfrak L\), and \(F\) be the corresponding subfield of \(L\). The lifting \(\lambda\) of the Galois 2-cohomology group \(H^2(\mathfrak L/\mathfrak H, F^*)\) \((F^*\)
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Local class field theory is treated by means of cohomology theory. Let \(L/K\) be a Galois extension with Galois group \(\mathfrak L\). Let \(\mathfrak H\) be an invariant subgroup of \(\mathfrak L\), and \(F\) be the corresponding subfield of \(L\). The lifting \(\lambda\) of the Galois 2-cohomology group \(H^2(\mathfrak L/\mathfrak H, F^*)\) \((F^*\)
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Galois Cohomology and Class Field Theory
Universitext, 2020 This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory.
David Harari
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The Structure of Local Class Field Theory
American Journal of Mathematics, 1938Ist für einen diskret-bewerteten perfekten Körper \(k\) der Restklassenkörper \(\mathfrak k\) endlich, so gelten die bekannten Sätze der lokalen Klassenkörpertheorie über \(k\): Für jedes \(n\) gibt es genau einen unverzweigten Erweiterungskörper vom Grade \(n\) über \(k\), und dieser ist zyklisch.
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