Find $\operatorname{Gal}(L/K)$, where $K=F_q(T)$ and $L$ is the spiliting field of $p(x)=Tx+x^q+T^q x^{q^2}$. 
Let $K=F_q(T)$ and $L$ be the splitting field of the polynomial $p(x)=Tx+x^q+T^q x^{q^2}$ over $K$. Find the Galois group $\operatorname{Gal}(L/K)$.

This is from homework of Drinfeld modules. Thanks for the help of my two classmates and the notes http://personal.psu.edu/mup17/Research/ODM.pdf, I know the Galois group of this polynomial $\text{Gal}(K(\phi[T])/K) $ is contained in $\text{GL}_2 (F_q) $ where $\phi[T]$ is the roots of $\phi_T (x)=Tx+x^q +T^q x^{q^2 -1}$ which is a $F_q$-module of rank two.(Refer to the notes http://personal.psu.edu/mup17/Research/ODM.pdf). In fact, this is contained in $\text{SL}_2 (F_q) $.
 A: Let $\alpha, \beta$ be a $F_q$-basis of $\phi[T]$. Then
$$
\phi_T(x)=T^{q} \prod_{m, n \in \mathbb{F}_{q}}(x+m \alpha+n \beta)=T^{q} \frac{\Delta(\alpha, \beta, x)}{\Delta(\alpha, \beta)}
$$
Comparing the coefficient of $x$-term we get $(T \Delta(\alpha, \beta))^{q-1}=1$. Thus $\Delta(\alpha, \beta)=m / T \in K$ for some $m \in F_q^\times.$ Then $\forall \sigma=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in \text{Gal}(K(\phi[T])/K) \subset \mathrm{GL}_{2}\left(F_q\right)$,
$$
\sigma(\Delta(\alpha, \beta))=\Delta(\sigma(\alpha, \beta))=\Delta(a \alpha+b \beta, c \alpha+d \beta)=\operatorname{det}(\sigma) \Delta(\alpha, \beta) .
$$Since $\Delta(\alpha, \beta) \in K$, then $\text{det}(\sigma)=1$ and hence $\sigma \in \text{SL}_2 (F_q) $. Next, using the following fact from the notes http://personal.psu.edu/mup17/Research/ODM.pdf.
Let $\phi$ be a Drinfeld module of rank $r$ over $K$. Note that we can write $\phi_{T}(x)=x f\left(x^{q-1}\right)$, where $f(x) \in K[x]$ is a polynomial of degree $\left(q^{r}-1\right) /(q-1)$. Show that the splitting field of $f(x)$ is the subfield of $K(\phi[T])$ fixed by $\pi_{T}\left(G_{K}\right) \cap Z\left(F_q\right)$, where $Z\left(F_{q}\right)$ denotes the center of $\mathrm{GL}_{r}\left(F_q\right)$.
In this case, $\pi_T (G_K) \subset \text{SL}_2 (F_q) $ and then the order of $\pi_{T}\left(G_{K}\right) \cap Z\left(F_q\right)$ is less than 2 and $f(y)=T^2 +y+y^{q+1}$. So we know $[K(\phi[T]):K_f ] \le 2$.  Let $S=T^2$, then $f(y)=S+y+y^{q+1}$. Let L be the spliting field of $f$ over $F_q (S)$. This is the case of this question Computing the Galois group of the splitting field of $X^{q+1} + X + T$ over the function field $\mathbb{F}_q(T)$. So we know $[L:F_q (S)]= q(q+1)(q-1)$. Then $[K_f :K]\ge \frac{q(q-1)(q+1)}{2}$. So $Gal(K(\phi[T])/K)$ is the normal subgroup of $\text{SL}_2 (F_q)$ of index at most 2. But $\mathrm{SL}_{2}\left(F_q\right)$ has no normal subgroup of index two because its commutator subgroup is itself. Then $\text{Gal}(K(\phi[T])/K)=\mathrm{SL}_{2}\left(F_q\right)$.
