First I define the level of field. The level of a field $\mathbb K$ is the least $n$ such that $−1$ is a sum of $n$ squares in field, and is denoted by $S(\mathbb K)$. I know that the level of $\mathbb Q_2$ (the field of $2$-adic numbers) is $4$. It is an easy calculation:
You can see that every $2$-adic integer which is congruent to $1 \bmod 8$ is a square. So $-7$ is a square and hence $S(\mathbb K) \leqslant 4$. On other hand any integral square in $\mathbb{Q}_{2}$ is congruent to $0$, $1$, or $4 \bmod 8$ so $S(\mathbb K) = 4$.
I am looking for the level of $p$-adic number field. Any suggestions? Thanks.