Exercise 18, § 10

Let K be a field and F the field of rational fractions K(T). The group GL₂(K) operates on F by the law (yf)(T) = f $\left( \frac{aT + b}{cT + d} \right)$, for y = $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ ∈ GL₂(K) and f ∈ F. The centre K* . I of GL₂(K) operates trivially, whence we obtain, by passage to quotients, a homomorphism of PGL₂(K) = GL₂(K)/K* . I into the automorphism group of the extension F of K. This homomorphism is bijective (V, p. 148, Ex. 11). a) Let H be a subgroup of PGL₂(K) of finite order h, and let E be the field of invariants of H, so that F = E(T) is a Galois extension of E with Galois group H (V, p. 66, Th. 3). Let

be the minimal polynomial of T over E. For each i such that 0 ≤ i ≤ h, either S_i(T) ∈ K or E = K(S_i(T)). (Use Ex. 11, a) of V, p. 157.) b) For each s ∈ H let $\begin{pmatrix} a_s & b_s \\ c_s & d_s \end{pmatrix}$ be a representative in GL₂(K) and let H₀ be the subgroup of H consisting of those s for which cₛ = 0; let h₀ = Card H₀ and put R^H(T) = S_{h₀}(T). Show that E = K(R^H(T)) and that R^H(T) = P(T)/Q(T)^{h₀} where P is a polynomial of degree h and Q(T) = $\prod_s (c_s T + d_s)$ where s runs over a system of representatives of the cosets H₀s different from H₀; the degree of Q is thus (h/h₀) − 1. c) Suppose that K* contains a subgroup μₙ of finite order n. Let C, be the image in PGL₂(K) of the subgroup of GL₂(K) consisting of the matrices $\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix}$, a ∈ μₙ. Then C, (X) = Xⁿ − Tⁿ and R^{Cₙ}(T) = (−1)^{n+1} Tⁿ. d) Let w be the element of PGL₂(K) represented by $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. Then w² = 1 and wsw⁻¹ = s⁻¹ if s is represented by a diagonal matrix. The subgroup Dₙ of PGL₂(K) generated by Cₙ and w is therefore of order 2n. We have Dₙ(X) = (Xⁿ − Tⁿ) $\left( X^n - \left( \frac{1}{T} \right)^n \right)$ = X^{2n} + (−1) R^{Dₙ}(T) + 1 where R^{Dₙ}(T) = − ((−T)ⁿ + (−T)⁻ⁿ).

e) Let S be the group of order 3 in PGL_2(K) generated by the image of the matrix $\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$. We have $S(X) = (X - T) \left( X - \frac{1}{T-1} \right) \left( X - \frac{T+1}{T-1} \right)$ and $R^s(T) = \frac{T^3 - 3T + 1}{T^2 - 1}$

f) Suppose that K is a finite field with q elements. Let U be the image in PGL_2(K) of the group of all matrices $\begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}$, $b \in K$. Then

Let B be the group generated by $C_{q-1}$ and U; it is of order q(q - 1). We have