$p$-adics, elements of $\mathbb{Q}_5^\times/(\mathbb{Q}_5^\times)^2$? Here is a question surrounding the $p$-adics. I am curious as to what the description of the quotient group $\mathbb{Q}_5^\times/(\mathbb{Q}_5^\times)^2$ is, i.e. what are its elements?
Here is an idea. I know that there is an isomorphism of groups $\mathbb{Q}_p^\times/\mathbb{Z}_p^\times = \mathbb{Z}$ which sends the class of $p^n$ ($n \in \mathbb{Z}$) to $n$, if that helps.
 A: $\mathbb{Q}_5^{\times}/(\mathbb{Q}_5^{\times})^2\simeq \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, with one set of representatives being $\{1,2,5,10\}$. I chose $2$ because it's a quadratic non-residue mod 5.
This is a consequence of the fact that (1) $\mathbb{Q}_p^{\times}\simeq \mathbb{Z}\times\mathbb{Z}_p^{\times}$, (2) $\mathbb{Z}_5^{\times}\simeq \Big(\mathbb{Z}/5\mathbb{Z}\Big)^{\times}\times(1+5\mathbb{Z}_5)$, and (3) $1+5\mathbb{Z}_5\simeq \mathbb{Z}_5$, together with the fact that $2\mathbb{Z}_5=\mathbb{Z}_5$.
A: The question can be generalized, as well as the answer. Consider an extension K/$\mathbf Q_p$ , of degree $n$. The structure of the group K* is known. Fixing an uniformizer $\pi $ of K , we have :     K* $\cong$ $(\pi) \times W \times U^1$, where $W$ is the group of roots of 1 contained in K*  , coming from Hensel’s lemma, and $U^1$ is the group of elements congruent to 1 mod $\pi$, isomorphic as a $\mathbf Z_p$-module to the direct product of  $n$ copies of $\mathbf Z_p$ and the group $\mu$ of $p$-primary roots of 1 in K* (see e.g. Serre’s « Local Fields », chapter 14, propos. 10) . This gives the structure of $K^*/K^{*n}$ for any integer $n$. For instance, take $n$ to be a prime $q$ and $p$ odd. Then $K^*/K^{*q}$ can be viewed as a vector space over $\mathbf F_q$ , of dimension equal to 1 + $\delta$ if $p$ $\neq$$q$ , n + 1 + $\delta$ if $p = q$, with $\delta$ = 1 (resp. 0) if K contains (resp. does not contain) a primitive q-th root of 1 .
