Introduction to Global Class Field Theory
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Global class field theory for number fields contains, roughly speaking, two classes of largely interdependent results: (A) Results showing that the classical invariants of the base field K alone (generalized class groups, unit groups) are sufficient to describe the Galois ab group of the abelian closure K of K and the decomposition law of places of K in its subextensions. (B) Results which illustrate the local-global principle, coming from the existence of the global reciprocity law which is (correctly) considered as a monumental generalization of Gauss’s quadratic reciprocity law 1 , and which rely on normic aspects of class field theory, developed essentially by Takagi, Artin, and Hasse. The complexity of the use of class field theory comes not only from the multiplicity of the above viewpoints, but also from the fact that it is often necessary to use nonalgebraic arguments (which are, in fact, independent from class field theory since, as Chevalley has shown in [h, Che2, § 9], it is possible to avoid any analytical argument in the proofs). There are two aspects of this, as follows. (C) The construction of abelian number fields, whose existence is assured by (A); this is the starting point for “Kronecker’s Jugendtraum” (for an imaginary quadratic base field), or more generally Hilbert’s twelfth problem whose aim is to extend to an arbitrary base field K the theory of cyclotomic fields over Q. This aspect can be studied using three fundamentally different paths: (α) Stark’s conjectures, described in [TBS]: these are analytical conjectures linked to classical Artin L-functions of number fields. They allow a numerical approach to the construction of ray class fields; 2 (β) arithmetic geometry: it shows that certain algebraic numbers coming from torsion points of curves or algebraic varieties also give constructions of abelian extensions; this quite old aspect has its origins in complex multiplication, and has had numerous theoretical and numerical developments; 3 1 2 3
and a few others, the history of which is described in [f, Lem]. See [e, Ko3, Ch. 5, § 1], then [Ro], [j, Coh2, Ch. 6, §§ 1, 2], and [CohR]. See [Deu], [d, CF, Ch. XIII], then [e, Ko3, Ch. 2, § 2], [j, Coh2, Ch. 6, § 3].
G. Gras, Class Field Theory © Springer-Verlag Berlin Heidelberg 2003
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Introduction to Global Class Field Theory
(γ) classical Kummer theory: this theory cannot be separated from class field theory since it is the tool used to prove the existence theorems of (A). It allows a direct numerical approach, however. 4 Indeed, in the extension K of K obtained by adjoining suitable roots of unity, we must determine a suitable radical which presupposes the knowledge of the class and unit group of K , which is possible only in terms of classical geometry of numbers. We can then come back down to K and give explicitly a polynomial defining the desired abelian extension of K which had been characterized in terms of class field. ˇ (D) Finally, density theorems such as the Cebotarev density theorem on Frobenius’ automorphisms, which for K = Q is simply the
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