homomorphism from $F^\times \times F^\times$ to Aut$(F)$

Question

Let $F$ be a field. Then $(F;+)$ and $(F^\times; \cdot)$ are groups. Let $$A = \{(a_{ij})_{2 \times 2}\in GL_2(F)|a_{ii}=1 \ \ \forall \ i=1,2 \ \mbox{and} \ a_{21} = 0\}$$ and $$B = \{\mbox{diagonal matrix in} \ GL_2(F)\}.$$ Then $A$ is isomorphic to $F$ and $D$ is isomorphic to $F^\times \times F^\times$.

Describe the homomorphism from $B$ to Aut$(A)$ in term of these isomorphism. (isomorphism ? does it refer to isomorphisms between $A$ and $F$, or between $D$ and $F^\times \times F^\times$ ? Or the automorphism in Aut$(F)$ ?)

(Hint : That is, show that each element of $F^\times \times F^\times$ acts as an automorphism on $F$)

I do not understand the hint, and how to do this type of problem.

So I guess assume that I can define an action of the group $F^\times \times F^\times$ on the group $F$. Then the action induces for each element $(\alpha, \beta)$ in $F^\times \times F^\times$, an element in the permutation group on $F$. Since the problem is about automorphism group, I think it might be possible to create the action to induce not just element in permutation on $F$, but an element in Aut$(F)$ ?

If that such action exists, then the permutation representation of $F^\times \times F^\times$ (homomorphism from $F^\times \times F^\times$ to the permutation group on $F$ which in this case should be the Aut$(F)$) is the relation I can use to describe ?

Any suggestion how to begin ? Is my idea, more or less, the right way to solve the problem ?

Answer 1

The exercise wanted you to consider first an action of matrix group $B$ on $A$, i.e. a homomorphism $h:B\to Aut(A)$.

Given by conjugation: $h(D):=\,M\mapsto DMD^{-1}\ \in Aut(A)\ $ for $D\in B$.

Then, use the isomorphisms $\varphi:A\overset\cong\to F^+$ and $\psi:B\overset\cong\to F^\times\times F^\times$ to create the required action of $F^\times\times F^\times$ on $F^+$.

These are $2\times 2$ matrices, even with lots of constraints: it's an easy calculation to find out the exact automorphism of $F^+$ that is assigned to a given $(a,b)\in F^\times\times F^\times$.