I'd like to find if possible the orders possible for $1-\gamma$ given a $\gamma\in\mathbb{F}_q$ of given order $\mathcal{o}$ (where $q=p^f$).

Which is clear is that these order have the same degree (degree: minimal $f$ such that $\mathcal{o}\mid p^f-1$ ie the degree of the minimal polynomial of $\gamma$ ie the minimal $f$ such that $\gamma$ is in $\mathbb{F}_{p^f}$).

More precisely if $\mu$ is the minimal polynomial of $\gamma$ on $\mathbb{F}_p$ the minimal polynomial of $1-\gamma$ is $\mu(1-X)$. So it only depends of the conjugacy class of $\gamma$ (that can be seen with Frobenius too).

For example with $\gamma\in\mathbb{F}_9\setminus\mathbb{F}_3$:

• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,1}=\mu=X^2-X-1$ have $1-\gamma$ of same order because $\mu(1-X)=\mu(X)$;
• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,2}=\mu=X^2+X-1$ have $1-\gamma$ of order 4 because $\mu(1-X)=X^2+1=\Phi_4(X)$;
• the $\gamma$ of order 4 with minimal polynomial $\Phi_4=X^2+1$ have $1-\gamma$ of order 8 because for involutive reasons $\Phi_4(1-X)=\Phi_{8,2}(X)$.

As we see here there is more generally an action of the affine transformation (and even of $\operatorname{Gl}_2(\mathbb{F}_q)$) on the conjugacy class of $\mathbb{F}_q$ elements which respect the orders.

I have seen on the net studies of the impact on the order of $\gamma\mapsto\gamma+1/\gamma$ (order of related elements) but I can't find anything on the impact of $\gamma\mapsto1-\gamma$.

I'd like to find if possible the orders possible for $1-\gamma$ given a $\gamma\in\mathbb{F}_q$ of given order $\mathcal{o}$ (where $q=p^f$).

Which is clear is that these order have the same degree (degree: minimal $f$ such that $\mathcal{o}\mid p^f-1$ ie the degree of the minimal polynomial of $\gamma$ ie the minimal $f$ such that $\gamma$ is in $\mathbb{F}_{p^f}$).

More precisely if $\mu$ is the minimal polynomial of $\gamma$ on $\mathbb{F}_p$ the minimal polynomial of $1-\gamma$ is $\mu(1-X)$. So it only depends of the conjugacy class of $\gamma$ (that can be seen with Frobenius too).

For example with $\gamma\in\mathbb{F}_9\setminus\mathbb{F}_3$:

• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,1}=\mu=X^2-X-1$ have $1-\gamma$ of same order because $\mu(1-X)=\mu(X)$;
• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,2}=\mu=X^2+X-1$ have $1-\gamma$ of order 4 because $\mu(1-X)=X^2+1=\Phi_4(X)$;
• the $\gamma$ of order 4 with minimal polynomial $\Phi_4=X^2+1$ have $1-\gamma$ of order 8 because for involutive reasons $\Phi_4(1-X)=\Phi_{8,2}(X)$.

As we see here there is more generally an action of the affine transformation (and even of $\operatorname{Gl}_2(\mathbb{F}_q)$) on the conjugacy class of $\mathbb{F}_q$ elements which respect the degree.

I have seen on the net studies of the impact on the order of $\gamma\mapsto\gamma+1/\gamma$ (order of related elements) but I can't find anything on the impact of $\gamma\mapsto1-\gamma$.

I'd like to find if possible the orders possible for $1-\gamma$ given a $\gamma\in\mathbb{F}_q$ of given order $\mathcal{o}$ (where $q=p^f$).

Which is clear is that these order have the same degree (degree: minimal $f$ such that $\mathcal{o}|p^f-1$ ie the degree of the minimal polynomial of $\gamma$ ie the minimal $f$ such that $\gamma$ is in $\mathbb{F}_{p^f}$).

More precisely if $\mu$ is the minimal polynomial of $\gamma$ on $\mathbb{F}_p$ the minimal polynomial of $1-\gamma$ is $\mu(1-X)$. So it only depends of the conjugacy class of $\gamma$ (that can be seen with Frobenius to).

For example with $\gamma\in\mathbb{F}_9\setminus\mathbb{F}_3$:

• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,1}=\mu=X^2-X-1$ have $1-\gamma$ of same order because $\mu(1-X)=\mu(X)$;
• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,2}=\mu=X^2+x-1$ have $1-\gamma$ of order 4 because $\mu(1-X)=X^2+1=\Phi_4(X)$
• the $\gamma$ of order 4 with minimal polynomial $\Phi_4=X^2+1$ have $1-\gamma$ of order 8 because for involutive reasons $\Phi_4(1-X)=\Phi_{8,2}$

As we see here there is more generaly an action of the affine transformation (and even of $\text{Gl}_2(\mathbb{F}_q)$) on the conjugacy class of $\mathbb{F}_q$ elements which respect the orders.

I have seen on the net studies of the impact on the order of $\gamma\mapsto\gamma+1/\gamma$ (order of related elements) but I can't find nothing on the impact of $\gamma\mapsto1-\gamma$.

Thanks for your help

I'd like to find if possible the orders possible for $1-\gamma$ given a $\gamma\in\mathbb{F}_q$ of given order $\mathcal{o}$ (where $q=p^f$).

Which is clear is that these order have the same degree (degree: minimal $f$ such that $\mathcal{o}\mid p^f-1$ ie the degree of the minimal polynomial of $\gamma$ ie the minimal $f$ such that $\gamma$ is in $\mathbb{F}_{p^f}$).

More precisely if $\mu$ is the minimal polynomial of $\gamma$ on $\mathbb{F}_p$ the minimal polynomial of $1-\gamma$ is $\mu(1-X)$. So it only depends of the conjugacy class of $\gamma$ (that can be seen with Frobenius too).

For example with $\gamma\in\mathbb{F}_9\setminus\mathbb{F}_3$:

• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,1}=\mu=X^2-X-1$ have $1-\gamma$ of same order because $\mu(1-X)=\mu(X)$;
• the $\gamma$ of order 8 with minimal polynomial $\Phi_{8,2}=\mu=X^2+X-1$ have $1-\gamma$ of order 4 because $\mu(1-X)=X^2+1=\Phi_4(X)$;
• the $\gamma$ of order 4 with minimal polynomial $\Phi_4=X^2+1$ have $1-\gamma$ of order 8 because for involutive reasons $\Phi_4(1-X)=\Phi_{8,2}(X)$.

As we see here there is more generally an action of the affine transformation (and even of $\operatorname{Gl}_2(\mathbb{F}_q)$) on the conjugacy class of $\mathbb{F}_q$ elements which respect the orders.

I have seen on the net studies of the impact on the order of $\gamma\mapsto\gamma+1/\gamma$ (order of related elements) but I can't find anything on the impact of $\gamma\mapsto1-\gamma$.