I asked same question on MSE before, but I didn't get any answer yet.

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.

More precisely, we can do analysis on $\mathbb{Q}_{p}$. We have an exponential and logarithm function on $\mathbb{Q}_{p}$ (at least for $p>2$), and we understand unit group $\mathbb{Z}_{p}^{\times}$ well, etc. But I don't know how to prove it in a purely local and analytic way.