I just read the proof of the quadratic reciprocity from the Artin reciprocity here Eisenstein and Quadratic Reciprocity as a consequence of Artin Reciprocity, and Composition of Reciprocity Laws given by Ted. It looks great but I cannot see where Artin reciprocity shows up in the proof. Probably I am missing something silly there. Can anyone remind me where it is used?
2026-03-25 14:26:08.1774448768
Proof of quadratic reciprocity from Artin reciprocity
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See Example 6.5 here. Artin reciprocity is used to interpret the Legendre symbol $(\mathbf Z/p\mathbf Z)^\times \to \{\pm 1\}$ for an odd prime $p$ as the Artin map from a certain generalized ideal class group of $\mathbf Q$ to the Galois group of a certain quadratic extension of $\mathbf Q$.