Class Field Theory
Class field theory asks how the abelian extensions of a field can be described using the field itself. For a nonarchimedean local field, reciprocity identifies Galois groups with quotients of the multiplicative group by norm subgroups, and the existence theorem says which subgroups arise. For number fields, ray class groups describe Frobenius automorphisms, while the idèle class group gives a single description of all finite abelian extensions.
These descriptions also constrain ramification, determine the least ray class modulus needed for an extension, and relate global norm equations to local tests. Over the rational numbers, the Kronecker–Weber theorem makes the extensions concrete by placing them in cyclotomic fields. The global results here concern number fields, not global function fields. The local reciprocity map and Hilbert pairing results assert existence without imposing additional normalization.
Selected theorems and definitions
- theorem
Finite local reciprocity
A finite abelian extension is recovered as a quotient of the base field's multiplicative group by its norm subgroup.
- theorem
Profinite local reciprocity
All finite abelian local extensions fit into one topological isomorphism.
- theorem
The local existence correspondence
Every open finite-index subgroup is a norm subgroup, with the correct reversal of inclusion.
- theorem
Finite global reciprocity
A ray class Artin map realizes Frobenius automorphisms in a finite abelian number-field extension.
- theorem
Topological global reciprocity
The identity component is exactly the part removed from the idèle class group to obtain the abelian Galois group.
- theorem
The Hasse–Arf theorem
Upper ramification breaks of a finite abelian local extension are integral.
- theorem
A perfect local Hilbert pairing
Power classes are dual to their roots-of-unity-valued characters.
- theorem
The cyclic Hasse norm theorem
For a cyclic number-field extension, local norm equations have a global solution.
- theorem
The conductor of an abelian extension
A unique least modulus measures which ray class fields contain the extension.
- theorem
The Kronecker–Weber theorem
Every finite abelian extension of the rational numbers embeds into a cyclotomic field.
References used in these guides
- Jean-Pierre Serre. Local Fields. Springer, New York, Graduate Texts in Mathematics 67, 1979. English translation of Corps locaux; local reciprocity, ramification, local symbols, and the existence theorem.
- J. S. Milne. Class Field Theory, version 4.03. 2020. Author's notes, dated 6 August 2020. Chapters I, III, V, VII, and VIII give complementary formulations and historical notes.
- Jonathan Lubin and John Tate. Formal Complex Multiplication in Local Fields. Annals of Mathematics, Second Series, 81 (2), 380–387, 1965. Original paper on the explicit formal-group construction of local abelian extensions; included as historical context, not as a claim that every proof below follows this construction.
- Cahit Arf. Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper. Journal für die reine und angewandte Mathematik, 181, 1–44, 1940. The publisher records the print year as 1940; the later online publication date is not the original publication year.