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.
Asked
Active
Viewed 404 times
12
J. W. Tanner
- 64,851
- 5
- 45
- 89
Seewoo Lee
- 15,748
-
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 '19 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 '19 at 16:48
-
Might be worth asking on MSO with a link back to here : https://mathoverflow.net/ – Martin Hansen Apr 03 '19 at 22:39
-
2Now posted also on MathOverflow: Is there a proof of quadratic reciprocity using $p$-adic numbers? – Martin Sleziak Apr 04 '19 at 05:59