Class Field Theory

Abelian extensions of number fields, the Artin map, and the structure of Galois groups via class groups

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Artin map: Cl(K) ≅ Gal(H/K), p ↦ Frob_p — reciprocity law

The Hilbert class field H of K is the maximal unramified abelian extension, with Gal(H/K) ≅ Cl(K). For ℚ(√-163), h=1 so ℚ(√-163) is its own Hilbert class field — one of 9 Heegner numbers. Primes split/remain inert/ramify in K based on the Legendre symbol (d/p).