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The exercise is as follows:

Let $\sigma$, $\tau \in Aut(\mathbb{F}_5 (X) / \mathbb{F}_5 )$, where $\sigma (X) = X+1$ and $\tau (X) = \frac{1}{X}$. ($X$ is the symbol or variable.) Let H be the group generated by $\sigma, \tau$. Compute the order of $H$.

I know that $\sigma^5 = 1$ and $\tau ^ 2 = 1$, so $10 \mid |H|$, and $[\mathbb{F}_5 (X):\mathbb{F}_5 (X)^H] = |H|$.But how to compute $\mathbb{F}_5 (X)^H$? I appreciate any help you can provide.

(My textbook is J.S.Milne's Fields and Galois Theory, and I have learned all the book this semester except the eighth chapter. )

shwsq
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    Hint: Those are both so called fractional linear transformation, i.e. transformations of the type $$X\mapsto \frac{aX+b}{cX+d},$$ where, suggestively, $ad-bc\neq0$. Composing them is not unlike multiplying matrices of the form $$\pmatrix{a&b\cr c&d \cr}.$$ Do observe that ALL scalar matrices yield the identity automorphism. That way you get a tool for calculating the order of the group. – Jyrki Lahtonen Jun 06 '24 at 07:25
  • @Jyrki Lahtonen Thank you! Now I can calculate the order by enumeration but it is very complex. Can you give an answer? – shwsq Jun 06 '24 at 09:03
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    Do you know the cardinality of $GL_k(\Bbb F_{p^m})$? Then, as Jyrki says, $H$ would be isomorphic to the quotient of $GL_2(\Bbb F_5)$ by its center (why is the map bijective?) – Amateur_Algebraist Jun 06 '24 at 20:53
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    Can you think of polynomials fixed by $\sigma$? What about rational functions fixed by $\tau$? – Steve D Jun 08 '24 at 14:54
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    Yes, but I cannot find a rational function fixed by both $\sigma$ and $\tau$. Could you explain what to do next after finding $x^5-x$ and $x+\frac{1}{x}$? – shwsq Jun 08 '24 at 23:25
  • You should think about what a general rational function fixed by $\sigma$ or $\tau$ would look like. For example, what does it mean for $$\tau(p(x))/\tau(q(x)) = p(x)/q(x)$$ – Steve D Jun 09 '24 at 00:34
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    Finding the fixed field, while easy for the cyclic subgroups of orders five and two (you got those!), may be a bit harrowing, perhaps Steve D knows something I don't :-). Anyway, calculating the order of the generated group should not be too difficult. Observe that the determinants of the matrices corresponding to $\sigma$ and $\tau$ are $\pm1$, so you won't get all of $GL_2(\Bbb{F}_5)$. – Jyrki Lahtonen Jun 09 '24 at 04:29
  • (cont'd) My main point is that you may need to take another look at texts on group theory at about this point. Many finer examples of Galois theory require a bit more familiarity with groups that was not covered in the introductory course. Solvability, simplicity of $A_n, n\ge5$, and then a few things about linear groups over finite fields. IIRC Milne has lecture notes on groups as well? – Jyrki Lahtonen Jun 09 '24 at 04:31

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