# 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

 $\mathrm{GL}_{n}(k[t])\cong\mathrm{GL}_{n}(k)*_{B(k)}B(k[t]).$
Title Nagao’s theorem NagaosTheorem 2013-03-22 13:58:31 2013-03-22 13:58:31 bwebste (988) bwebste (988) 6 bwebste (988) Theorem msc 20G15