Is There an Iff Relation for Polynomial Divisibility in the Ring F[t, t^-1]?

[mathmari]

Hey!

We have the following lemma:

Assume that the characteristic of $F$ is $p$ and $p>2$.
Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$) if and only if $(\exists s \in \mathbb{Z}) m=np^s$.
Can we say something about $p=2$ ?

$$(\exists s \in \mathbb{Z})m=2^sn \Leftrightarrow \dots$$

If $(\exists s \in \mathbb{Z})m=2^sn$ then we have that:
$$t^m=t^{2^sn}=\left (t^n\right )^{2^s} \Rightarrow t^m-1=\left (t^n\right )^{2^s}-1=\left (t^n-1\right )^{2^s}$$

Can we write with that a $\Leftrightarrow$ relation?

 

[mathmari]

There are the following lemmas:

Lemma 1.

For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and $t-1$ divides $x-1$ (the divisibilities are meant, of course, in $F[t, t^{-1}]$).
Lemma 2.

$t^n-1$ divides $t^m-1$ in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$) if and only if $n$ divides $m$ in $\mathbb{Z}$.

If $\exists s \in \mathbb{Z}$ so that $m=2^s n$, then $2 \mid m$ and $n \mid m$.

We have the following:

$$2 \mid m \Leftrightarrow t^2-1 \mid t^m-1$$

and $$n \mid m \Leftrightarrow t^n-1 \mid t^m-1$$

Is this an "iff" statement?