Let $K$ be a local field of characteristic zero. Is it true that $K$ is isomorphic to the completion of a number field under some valuations? If yes, then how to prove it?

I ask this question since in a paper it is said that $k_v^*/(k_v^*)^2$ where $k_v$ is the completion of a number field at some place $v$ has order 1, 2, 4 or 8. From the structure theory of local fields this order argument is incorrect for general local fields of characteristic zero.

If not every local field of characteristic zero comes from the completion of some number field and the above claim about the order of $k_v^*/(k_v^*)^2$ is correct. Can some one prove it or give a reference for it?

Let $K$ be a local field (locally compact topological field) of characteristic zero. Is it true that $K$ is isomorphic to the completion of a number field under some valuations? If yes, then how to prove it?

I ask this question since in a paper it is said that $k_v^*/(k_v^*)^2$ where $k_v$ is the completion of a number field at some place $v$ has order 1, 2, 4 or 8. From the structure theory of local fields this order argument is incorrect for general local fields of characteristic zero.

If not every local field of characteristic zero comes from the completion of some number field and the above claim about the order of $k_v^*/(k_v^*)^2$ is correct. Can some one prove it or give a reference for it?