From [Wikipedia][1]: given $a\in K^\times$,
`(a,b)=1 for all b [in K*] if and only if a is in K*ⁿ`

So suppose that $(\frac{a\ ,\ K^\times\!}{p})\neq 1$ [$p$ here $:=$ prime ideal generated by $n$ above].  Given $h\in\Bbb N$, are there any hypotheses that would allow us to conclude that $(\frac{a\ ,\ b}{p})\neq 1$ for some $b\in U_p(h)$? (i.e. so $b-1\in p^h$)

This seems like should be trivially true, at least for $h=0$; however, I cannot think of a reason why.


  [1]: https://en.wikipedia.org/wiki/Hilbert_symbol#Properties_2