Showing that a map $\mathrm{GL}(2,k) \to \mathrm{Gal}(k(x)/k)$ is a group homomorphism

Question

This is Problem 15 from section 2 on page 26 of Patrick Morandi's Field and Galois Theory.

Let $k$ be a field and $k(x)$ be the rational function field in one variable over $k$. Show that the map from the set of invertible $2 \times 2$ matrices over $k$ to $\mathrm{Gal}(k(x)/k)$ given by $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \varphi, $$ where $\varphi(x) = (ax+b)/(cx+d)$, is a group homomorphism.

However, my calculation shows that it must be an anti-homomorphism. I have given my work below.

I have already shown that every element $\varphi \in \mathrm{Gal}(k(x)/k)$ satisfies $\varphi(x) = (ax+b)/(cx+d)$ where $$ \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} \neq 0. $$ So, $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \varphi, $$ where $\varphi(x) = (ax+b)/(cx+d)$, is a well-defined function from $\mathrm{GL}(2,k)$ to $\mathrm{Gal}(k(x)/k)$. To show that it is a group homomorphism, I need to show that if $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \varphi \quad \text{and} \quad \begin{pmatrix} p & q \\ r & s \end{pmatrix} \mapsto \psi, $$ then $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} ap + br & aq + bs \\ cp + dr & cq + ds \end{pmatrix} \mapsto \varphi \circ \psi. $$ However, my computation shows me that $$ \begin{align} \varphi \circ \psi (x) &= \varphi \left( \frac{px+q}{rx+s} \right)\\ &= \frac{p\left(\frac{ax+b}{cx+d}\right)+q}{r\left(\frac{ax+b}{cx+d}\right)+s}\\ &= \frac{(pa+qc)x+(pb+qd)}{(ra+sc)x+(rb+sd)} \end{align}, $$ which is the image of $$ \begin{pmatrix} p & q \\ r & s \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix}. $$ Thus, the given map is actually an anti-homomorphism from $\mathrm{GL}(2,k)$ to $\mathrm{Gal}(k(x)/k)$.

Could someone verify my calculation and/or confirm whether this is a typo in the textbook? It isn't mentioned as such in the list of typos.

Answer 1

The map as defined in the question details is an anti-homormorphism and not a homomorphism. Therefore, the problem is not stated correctly in the textbook.

Answer 2

This is not a typo and your book is correct. This problem is "one of those things" and I think many people will relate to the confusion...

$\varphi \circ \psi$ is meant to be $$\varphi \circ \psi(x) = \frac{a \psi(x) + b}{c \psi(x) + d}.$$