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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?

Ja_1941
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    $p^$ is a square mod $q$ iff $q$ splits in $K=\mathbb{Q}(\sqrt{p^})$ iff the Frobenius $\sigma$ at $q$ of $C=\mathbb{Q}(\mu_p)$ is in $Gal(C/K)$. Now $Gal(C/\mathbb{Q})$ is cyclic and $Gal(C/K)$ has index two, so $Gal(C/K)=2Gal(C/\mathbb{Q})$. Thus $p^$ is a square mod $q$ iff $\sigma$ is a square in $Gal(C/\mathbb{Q})$. But $Gal(C/\mathbb{Q}) \cong \mathbb{F}_p^{\times}$ under an isomorphism mapping $\sigma$ to $q$, so $p^$ is a square mod $q$ iff $q$ is a square in $\mathbb{F}_p^{\times}$, QED. No Artin reciprocity here, just classical algebraic number theory. – Aphelli Feb 02 '23 at 19:24
  • You might also want to look at Theorems 8.11 and 8.12 of Cox – rogerl Feb 02 '23 at 23:34

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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 $\mathbf Q$.

KCd
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