Suppose that $F$ is a global field containing $l$-th roots of unity where $l$ is a prime integer and $l\neq\mbox{char}\ F$ and that $S$ is a finite set of places $v.$ Given some $t_v\in F_v(c^{1/l})$ where $c\in F$ and $c^{1/l}\notin F_v$ for all $v\in S$. Can we find an element $t\in F(c^{1/l})$ such that $t/t_v$ is local $l$-th power in $F_v(c^{1/l})$ at every $v\in S$, i.e. $t/t_v\in F_v^{l}(c^{1/l})$?
This question is a detail from a paper that I'm reading, "Relations Between K2 and Galois Cohomology" by Tate.
I'm trying to use the fact that $F(c^{1/l})$ is dense at every place $v\in S$ and that $t_vF_v^{\times l}(c^{1/l})$ is an open subset of $F_v^{\times}(c^{1/l}).$ But it only helps me find a $t$ that satisfies for only one place but not every place at $S.$
Can anyone helps me a little? Thanks.