Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $i,j$ where $1\le i,j<\ell$ such that

$\hspace{5cm} Tr_{\mathbb{F}_{q^\ell}/\mathbb{F}_{q}}(\zeta^{1-q^i})=0$

and

$\hspace{5cm}Tr_{\mathbb{F}_{q^\ell}/\mathbb{F}_{q}}(\zeta^{1-q^j})=0$