I want to prove that the group $$B_n = \{ \begin{pmatrix} a_1 & b_{1,2} & \cdots & b_{1,n} \\ & \ddots & & \vdots \\ & & a_{n-1} & b_{n-1, n} \\ & & & a_n \end{pmatrix} | b_{ij} \in F, a_i \in F^* \} $$
is solvable.
I though about using the homomorphism $f: B_n \rightarrow (F^*)^n$ , which is declared $$f(\begin{pmatrix} a_1 & b_{1,2} & \cdots & b_{1,n} \\ & \ddots & & \vdots \\ & & a_{n-1} & b_{n-1, n} \\ & & & a_n \end{pmatrix}) = (a_1,....,a_n)$$
this way, I get $$Kerf = \{ \begin{pmatrix} 1 & b_{1,2} & \cdots & b_{1,n} \\ & \ddots & & \vdots \\ & & 1 & b_{n-1, n} \\ & & & 1 \end{pmatrix} | b_{ij} \in F* \} $$
and since $Ker(f) \lhd B_n$ I thought about using the theorem which says that: if $N \lhd G$ is solvable and $G/N$ is solvable, then G is also solvable.
I also know that $Im(f) = (F^*)^n$.
I got stuck proving that $Ker(f)$ and $B_n/Ker(f)$ are solvable
Help would be appreciated
