Square classes p-adic numbers isomorphism I was reading about the Hilbert symbol and the Hasse-Minkowski theorem and found this statement in the book I was reading :
$ |\frac{Q_p^{*}}{(Q_p^{*})^{2}} |=2^r$; with $r=2$ for $p \neq 2$ and $r=3$ for $p=2$
I was thinking about proving it directly by showing an isomorpishm, but I got a bit stuck. I'd apreciate any advice or hint you could give me in order to prove it.
Thank you so much in advance.

It may be possible to prove this via Hensel's lemma, but I think a more persuasive argument (perhaps the most standard one) is by p-adic versions of exponential and logarithm (yes, taking p-adic inputs and producing p-adic outputs). The $n!$ in the denominator of the exponential is no longer large, p-adically, indeed, so this causes some trouble, rather than helping convergence. The specifics about the convergence (in most textbooks...) show, qualitatively, that for given $n$, anything sufficiently near $1$ has an $n$-th root. The details work out to give what you are trying to prove in the square-root/square case. (Among other sources, my notes on alg no. th. cover such things: http://www-users.math.umn.edu/~garrett/m/number_theory/.)
