independency of power of linear independent elements in a field extension

Question

$F \hookrightarrow E$ is a field extension, $a_1, a_2, \cdots, a_n$ is a set of $F$-linear independent elements in $E$, $m$ is an integer coprime to $[E:F]$, is $a_1^m, a_2^m, \cdots, a_n^m$ always $F$-linear independent?

Answer 1

Take $F = \mathbb{Q}, E = \mathbb{Q}(\omega)$ where $\omega$ is a primitive third root of unity, and $m = 3$. Then $\{ 1, \omega \}$ is linearly independent (and in fact a basis) but $\{ 1, \omega^3 = 1 \}$ is not. Similar examples exist using cyclotomic extensions $\mathbb{Q}(\zeta_n)$ for any $n$ such that $\gcd(n, \varphi(n)) = 1$ (in particular, all prime $n$; this is A003277).