Let $K = \mathbb C(X)$, $\sigma: X \mapsto \xi X$ and $\tau: X \mapsto X^{-1}$. Find number of elements in $G$ generated by $\sigma$ and $\tau$.

Asked Oct 28 '14 at 08:13

Active Oct 28 '14 at 09:27

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Let $K = \mathbb C(X)$ (denote the field of fractions of $\mathbb C[X]$) and consider the following automorphisms of $K$:

$\sigma: X \mapsto \xi X$ and $\tau: X \mapsto X^{-1}$, where $\xi$ denote the primitive root of $1$ of order $3$.

I want to check the relations in group $G$ generated by $\sigma$ and $\tau$:

$\sigma^3 = 1 = \tau^2$ and $\tau \sigma \tau = \sigma^{-1}$.

and find the number of elements in $G$.

I know $\sigma^3(X) = \sigma(\sigma(\sigma(X))) = \sigma(\sigma(\xi X)) = \sigma(\sigma(\xi)\sigma(X)) = \sigma(\sigma(\xi) \xi X))$.

But how can I determine that $\sigma(\xi) = \xi$ such that I can conclude $\sigma^3: X \mapsto X$ ?

I've the same problem regarding $\tau^2 = 1$.

edited Oct 28 '14 at 09:27

asked Oct 28 '14 at 08:13

Shuzheng

I think it is assumed that $\sigma$ and $\tau$ fix elements in $\mathbb C$? – awllower Oct 28 '14 at 08:30
@awllower i think so... unless otherwise stated... – Oct 28 '14 at 08:31
@awllower, I guess you are right, that's also my idea. But this is not stated in the problem formulation. I guess I will contact the author. – Shuzheng Oct 28 '14 at 08:35
Will it help if I say that $\mathbb C(X)$ denote the field of fractions of $\mathbb C[X]$ ? – Shuzheng Oct 28 '14 at 08:39
I think more contexts help in determining if $\sigma$ is assumed to fix $\mathbb C$ or not. – awllower Oct 28 '14 at 08:46

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