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Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$. Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$, where both $\mathrm{Gal}(F/k)$ and $\mathrm{Gal}(E/F)$ are nonabelian simple finite groups. We have a norm map $N\colon E\to F$. Choose $x\in F^*$ which is not a norm, i.e. is not contained in the image of $N$.

Now let $K/k$ be a finite solvable field extension in $\overline k$. Then $K\cap E=k$, and we have a norm map of composites $N_K\colon KE\to KF$.

Does there always exist a finite solvable field extension $K/k$ such that $x$ is contained in the image of the map $N_K$?

If $k$ is a number field, the answer is YES. For an arbitrary field $k$ I expect in general the answer NO, but I cannot construct a counter-example.

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Just to remark that if $[E:F]=n$ and $L$ is a solvable Galois extension of $F$ containing an $n$th root of $x$, then $L$ is disjoint from $E$ and $x$ is a norm from $LE$ down to $LF=L$. So for a counterexample one wants to make sure that $L$ (a solvable extension of a non-solvable extension of $k$) can't be written as $KF$ for $K/k$ solvable. – Kevin Buzzard May 16 '11 at 23:19

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