Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(F)$ the groups of $E,F$-rational point on $G={\rm GL}_{n}$. Two elements $g,h\in G(E)$ are called $\sigma$-conjugate if $g=x^{-1}hx^{\sigma}$ for some $x\in G(E)$, and $N(g)=gg^{\sigma}\cdots g^{\sigma^{\ell-1}}$ is called the norm of $x$.
Question. Is the norm map an injection from the set of $\sigma$-conjugacy classes in $G(E)$ into the set of conjugacy classes in $G(F)$ ?
This is the assertion of lemma 1.1 in ''Simple Algebra, Base Change, and the Advanced Theory of the Trace Formula" written by J. Arthur and L. Clozel. Their proof was very hard to understand for me.
