Let $p$ be a prime. Given a positive integer $n$, does there exist a $\beta$ in an extension of $F_p$ such that

1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a high extension)

2) The order of $\beta$ is at most $2^{poly(n)}$; ( $\beta$ has small order)

3) $\beta F_p + \beta^n F_p + \beta^{n^2} F_p + ... + \beta^{n^n} F_p \subseteq < \beta > $; (the subgroup generated by $\beta$ contains a linear space )

Thanks,

Qi