# Is there a proof of quadratic reciprocity using $p$-adic numbers?

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.

• 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? – Servaes Mar 31 at 13:56
• @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. – Seewoo Lee Mar 31 at 16:48
• Might be worth asking on MSO with a link back to here : mathoverflow.net – Martin Hansen Apr 3 at 22:39
• Now posted also on MathOverflow: Is there a proof of quadratic reciprocity using $p$-adic numbers? – Martin Sleziak Apr 4 at 5:59