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I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can get it by considering the cyclotomic extension of $\mathbb{Q}_{p}$. However, I want to know if there's any proof of quadratic reciprocity that doesn't use any stronger result, but only uses some properties of $p$-adic numbers. Thanks in advance.

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  • $\begingroup$ There are fine proofs through cyclotomic extensions of $\Bbb{Q}$, without any reference to $p$-adic numbers at all. Do you want to avoid (cyclotomic) field extensions, and how do you feel about proofs without $p$-adic numbers? $\endgroup$ – Servaes Mar 31 at 13:56
  • $\begingroup$ @Servaes I know some proofs that doesn't use $p$-adic numbers at all. One of the reason that $p$-adic numbers are useful is that we know much about its algebraic and analytic properties. I think there might be a way to use such tools. $\endgroup$ – Seewoo Lee Mar 31 at 16:48
  • $\begingroup$ Might be worth asking on MSO with a link back to here : mathoverflow.net $\endgroup$ – Martin Hansen Apr 3 at 22:39
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    $\begingroup$ Now posted also on MathOverflow: Is there a proof of quadratic reciprocity using $p$-adic numbers? $\endgroup$ – Martin Sleziak Apr 4 at 5:59

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