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Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One supposes that $\mathbb F_q$ is embed in $K$. One considers elements of $K$: $w_1,\cdots,w_s$. Assume there exists $b_1,\cdots,b_s$ in $K$ not all zero such that for every $n\in\mathbb N$, one has $$\sum_{i=1}^sb_iw^{q^n}_i=0\qquad(**)$$ A result from Moore asserts that the $w_1,\cdots,w_s$ are $\mathbb F_q$-linearly dependent. Now, assume that $\overline{\mathbb F_q}$ is embed in $K$ and ($**$) holds only for an infinite increasing sequence $(n_k)_k$ of natural integers. My question: can one assert that the $w_i$'s are $\mathbb F_{q^r}$-linearly dependent for a certain natural integer $r$? I checked the case $s=2$, and it is true. Thanks in advance for any answer.

Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One supposes that $\mathbb F_q$ is embedded in $K$. One considers elements of $K$: $w_1,\dotsc,w_s$. Assume there exists $b_1,\dotsc,b_s$ in $K$ not all zero such that for every $n\in\mathbb N$, one has $$\sum_{i=1}^sb_iw^{q^n}_i=0.\tag{**}\label{starstar}$$ A result from Moore asserts that the $w_1,\dotsc,w_s$ are $\mathbb F_q$-linearly dependent. Now, assume that $\overline{\mathbb F_q}$ is embedded in $K$ and \eqref{starstar} holds only for an infinite increasing sequence $(n_k)_k$ of natural integers. My question: can one assert that the $w_i$'s are $\mathbb F_{q^r}$-linearly dependent for a certain natural integer $r$? I checked the case $s=2$, and it is true.

Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One suppose that $\mathbb F_q$ is embed in $K$. One considers elements of $K$: $w_1,\cdots,w_s$. Assume there exists $b_1,\cdots,b_s$ in $K$ not all zero such that for every $n\in\mathbb N$, one has $$\sum_{i=1}^sb_iw^{q^n}_i=0\qquad(**)$$ A result from Moore asserts that the $w_1,\cdots,w_s$ are $\mathbb F_q$-linearly dependent. Now, assume that $\overline{\mathbb F_q}$ is embed in $K$ and ($**$) holds only for an infinite increasing sequence $(n_k)_k$ of natural integers. My question: can one assert that the $w_i$'s are $\mathbb F_{q^r}$-linearly dependent for a certain natural integer $r$? I checked the case $s=2$, and it is true. Thanks in advance for any answer.

Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One supposes that $\mathbb F_q$ is embed in $K$. One considers elements of $K$: $w_1,\cdots,w_s$. Assume there exists $b_1,\cdots,b_s$ in $K$ not all zero such that for every $n\in\mathbb N$, one has $$\sum_{i=1}^sb_iw^{q^n}_i=0\qquad(**)$$ A result from Moore asserts that the $w_1,\cdots,w_s$ are $\mathbb F_q$-linearly dependent. Now, assume that $\overline{\mathbb F_q}$ is embed in $K$ and ($**$) holds only for an infinite increasing sequence $(n_k)_k$ of natural integers. My question: can one assert that the $w_i$'s are $\mathbb F_{q^r}$-linearly dependent for a certain natural integer $r$? I checked the case $s=2$, and it is true. Thanks in advance for any answer.

Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One suppose that $\mathbb F_q$ is embed in $K$. One considers elements of $K$: $w_1,\cdots,w_s$. Assume there exists $b_1,\cdots,b_s$ in $K$ not all zero such that for every $n\in\mathbb N$, one has $$\sum_{i=1}^sb_iw^{q^n}_i=0\qquad(**)$$ A result from Moore asserts that the $w_1,\cdots,w_s$ are $\mathbb F_q$-linearly dependent. Now, assume that $\overline{\mathbb F_q}$ is embed in $K$ and ($**$) holds only for an infinite increasing sequence $(n_k)_k$ of natural integers. My question: can one assert that the $w_i$'s are $\mathbb F_{q^r}$-linearly dependent for a certain natural integer $r$?

Thanks in advance for any answer.

Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One suppose that $\mathbb F_q$ is embed in $K$. One considers elements of $K$: $w_1,\cdots,w_s$. Assume there exists $b_1,\cdots,b_s$ in $K$ not all zero such that for every $n\in\mathbb N$, one has $$\sum_{i=1}^sb_iw^{q^n}_i=0\qquad(**)$$ A result from Moore asserts that the $w_1,\cdots,w_s$ are $\mathbb F_q$-linearly dependent. Now, assume that $\overline{\mathbb F_q}$ is embed in $K$ and ($**$) holds only for an infinite increasing sequence $(n_k)_k$ of natural integers. My question: can one assert that the $w_i$'s are $\mathbb F_{q^r}$-linearly dependent for a certain natural integer $r$? I checked the case $s=2$, and it is true. Thanks in advance for any answer.