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I am trying to determine the invariant rings of all finite subgroups $\Gamma$ of $\text{GL}(1,\mathbb{C})=\mathbb{C}^\ast$. I know that since $\Gamma$ is finite, the invariant ring is finitely generated, there is $1$ algebraically independent invariant and the invariant ring has an algebra consisting of at most $1+|\Gamma|$ invariants of degree bounded above by $|\Gamma|$. I am not sure how to classify all the invariant rings however as finite groups of $\mathbb{C}^\ast$ come in various forms (I'm thinking of roots of unity here for example). Any help/hints would be appreciated.

Saegusa
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  • Apart from the groups of $n$-th roots of unity what others are there?? – ancient mathematician Oct 11 '21 at 15:02
  • I don't know, I just assumed there might be more. If I know this, I'm still not sure how to find the invariant subring however. – Saegusa Oct 11 '21 at 15:17
  • There is an excellent proof here that such subgroups are cyclic, and that means they are groups of $n$-rou. https://math.stackexchange.com/questions/59903/finite-subgroups-of-the-multiplicative-group-of-a-field-are-cyclic – ancient mathematician Oct 11 '21 at 16:53
  • Isn't the Molien series of $G$ cyclic of order $N$ not just $(1-t^N)^{-1}$? What have you actually tried? – ancient mathematician Oct 11 '21 at 16:58
  • Ah yes that makes sense! We haven't hit Molien series in my course yet so I must admit I don't know what that is. The only things I know are the results I stated above so I thought there would be a somewhat simple way of seeing this but it's eluding me. – Saegusa Oct 11 '21 at 17:03
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    Well there are very few (homogenous) polynomials of degree $k$ in one variable. So when is $x^k$ fixed by $x\mapsto \omega x$ where $\omega$ is the primitive $N$-th root of unity? This is NotMySubject so I am not going to write a solution. – ancient mathematician Oct 12 '21 at 06:28

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