I was wondering about the quadratic closure of the field of $p$-adic numbers $\mathbb{Q}_p$ ($p$ prime) exists. What is known about it? Is there any explicit description of such a quadratic closure? Does it occur naturally in number theory?
I was thinking about a construction in geometry, that I would like to "port" to the $p$-adic numbers, and I was led to the questions above.
For $p\ne 2$ $$K=\bigcup_n \Bbb{Q}_p(\zeta_{2^n},p^{1/2^n})=\bigcup_n \Bbb{Q}_p(\zeta_{p^{2^n}-1},p^{1/2^n})$$ contains the square root of all its elements.
Proof : the binomial series $(1+x)^{1/2}=\sum_{k=0}^\infty {1/2\choose k} x^k$ converges for $v(x)>0$