Nagao’s theorem

For any integral domain $k$, the group of $n\times n$ invertible matrices with coefficients in $k[t]$ is the amalgamated free product of invertible matrices over $k$ and invertible upper triangular matrices over $k[t]$, amalgamated over the upper triangular matrices of $k$. More compactly

$${\mathrm{GL}}_n(k[t])\cong {\mathrm{GL}}_n(k){*}_{B(k)}B(k[t]).$$