5 - Class Field Theory
Published online by Cambridge University Press: 05 June 2012
Summary
The classical theory
Let k be an algebraic number field and K a finite abelian extension of k. The objective of classical class field theory, which was largely achieved, was to describe the properties of K in terms of objects in k. The theory was first formulated in the 1890s, partly by Weber (following Kronecker) and partly by Hilbert (following Kummer); but one crucial component was only provided by Artin in 1927. The first proofs were given by Takagi in the 1920s; he used complicated group-theoretic arguments which we now know to belong to group cohomology — a subject which at that time had not been invented.
Let k be an algebraic number field with class number h > 1. It is straightforward to show that there are algebraic number fields K ⊃ k such that every ideal in k becomes principal in K. Is there a canonical way of choosing K, and what additional properties will the extension K/k have? Hilbert conjectured that there is just one such field K with the following additional properties:
(i) dK/k = (1), so that the extension is unramified at all finite places;
(ii) the extension is also unramified at all infinite places, so that the places of K above a real infinite place of k are all real;
(iii) K is abelian over k with Galois group isomorphic to the ideal class group Ik of k.
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- Information
- A Brief Guide to Algebraic Number Theory , pp. 98 - 116Publisher: Cambridge University PressPrint publication year: 2001