field extension with degree of 2

Question

I need to decide if this following statement is true or false: $\forall K \text{ field (not algebraically closed) } \exists L,K\leq L: [L:K]=2$

For finite fields this statement should be true, but I have no idea if this is true for all $K$.

Answer 1

It is false.

The complex constructible numbers are not algebraically closed, but they are quadratically closed (everything has a square root).