Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of basises of the following form. Let $$\vec{\beta} = (\beta_0,\beta_1,\ldots,\beta_{n-1})$$ be a basis of the space $_RS$ with the property that there exists $k\in\{1,2,\ldots,n-1\}$ such that $$ \vec{\beta'} = (\beta_0,\ldots,\beta_{k-1},h\beta_k,\ldots,h\beta_{n-1}) $$ is a basis of the space $_RS$.

It is easily to see that $N\geq (q^n-q)\prod_{j=1}^{n-1}(q^n-2q^{j}+q^{j-1})$. This estimation obtained from the case $k = n-1$

It is also easily to see, that the basis of the form $(e,h,\ldots, h^{n-1})$ has the desirable property. My another question is: is there another approches for construction such basises?

Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of bases of the following form. Let $$\vec{\beta} = (\beta_0,\beta_1,\ldots,\beta_{n-1})$$ be a basis of the space $_RS$ with the property that there exists $k\in\{1,2,\ldots,n-1\}$ such that $$ \vec{\beta'} = (\beta_0,\ldots,\beta_{k-1},h\beta_k,\ldots,h\beta_{n-1}) $$ is a basis of the space $_RS$.

It is easily to see that $N\geq (q^n-q)\prod_{j=1}^{n-1}(q^n-2q^{j}+q^{j-1})$. This estimation obtained from the case $k = n-1$

It is also easily to see, that the basis of the form $(e,h,\ldots, h^{n-1})$ has the desirable property. My another question is: is there another approches for construction such bases?