Let $L|K$ be an abelian extension of number fields. What is the significance of the fact that the artin map for an abelian extension is surjective?
2026-03-26 14:34:58.1774535698
What is the significance of the fact that the artin map for an abelian extension is surjective?
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The main goal of global CFT is to classify/describe the abelian extensions of a number field K using only parameters belonging to K. Artin's reciprocity law does this job, but only because the Artin map is surjective. Besides, note that, given a finite abelian extension $L/K$, if there exists an Artin map $\psi: C_K/NC_L \to G(L/K)$, then it must be an isomorphism (Cassels-Fröhlich, chapter VII, consequence 9.4).