Suppose $F$ is a number field, and $a, b$ are non-zero elements. Does there always exist $x \in F$ such that the norm residue symbols (=cup products) are $(a, x)= 0 = (x, b) \in H^2(F, \mathbb{F}_2)$ AND such that $x$ is linearly independent from $a, b$ in $F^* / (F^*)^2$ ?
When $F= \mathbb{Q}$, i believe the answer is yes, using Dirichlet's theorem on primes in arithmetic progression (I may be wrong). What about the general case?
I'm hoping that there's a relatively elementary answer.
