- #1
mathmari
Gold Member
MHB
- 5,049
- 7
Hey! ![Eek! :eek: :eek:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
I want to show 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 $(\exists s \in \mathbb{Z}) m=np^s$. I have done the following:
$\Leftarrow $ :
$$\frac{t^m-1}{t^n-1} \overset{ m=np^s }{ = } \frac{t^{np^s}-1}{t^n-1}=\frac{(t^n)^{p^s}-1}{t^n-1}=\frac{(t^n-1)((t^n)^{p^s-1}+\dots +1)}{t^n-1}=(t^n)^{p^s-1}+\dots +1$$
Is this correct so far?
How can we continue to conclude that $(t^m-1)/(t^n-1)$ is a square ? $\Rightarrow $ :
$$\frac{t^m-1}{t^n-1}=a^2 \Rightarrow t^m-1=a^2(t^n-1)$$
How can we continue?
I want to show 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 $(\exists s \in \mathbb{Z}) m=np^s$. I have done the following:
$\Leftarrow $ :
$$\frac{t^m-1}{t^n-1} \overset{ m=np^s }{ = } \frac{t^{np^s}-1}{t^n-1}=\frac{(t^n)^{p^s}-1}{t^n-1}=\frac{(t^n-1)((t^n)^{p^s-1}+\dots +1)}{t^n-1}=(t^n)^{p^s-1}+\dots +1$$
Is this correct so far?
How can we continue to conclude that $(t^m-1)/(t^n-1)$ is a square ? $\Rightarrow $ :
$$\frac{t^m-1}{t^n-1}=a^2 \Rightarrow t^m-1=a^2(t^n-1)$$
How can we continue?
Last edited by a moderator: