Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 2 / d

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- Solution d

Solution

The ideal-theoretic existence theorem of global class field theory gives an inclusion-reversing correspondence between finite abelian extensions

L / K

and congruence subgroups

P_{K} (m) \subseteq H \subseteq I_{K} (m) .

The corresponding field satisfies

H = ker (I_{K} (m) Art_{L / K} Gal (L / K)), I_{K} (m) / H ≅ Gal (L / K) .

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