Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 2 / a

An element $g\in G(k)$ is semisimple when its image in a faithful finite-dimensional representation is diagonalizable, and unipotent when that image has every eigenvalue equal to one. These conditions are independent of the faithful representation. The Jordan decomposition in an affine algebraic group is the unique factorizationwith $g_s$ semisimple and $g_u$ unipotent.

In $G={\mathrm{GL}}_2$, matrices with two distinct eigenvalues form a dense open subset and are semisimple, so semisimple elements are dense. Their full locus is not open: a scalar matrix is semisimple, but every neighbourhood of it contains a nontrivial Jordan block. The group also has nonidentity unipotent matrices.

For a contrasting example take $G={\mathbb{G}}_a\times {\mathbb{G}}_m$. Its semisimple locus is $\{0\}\times {\mathbb{G}}_m$, which is closed and not dense, while $(a,1)$ for $a\ne 0$ is unipotent and $(0,t)$ for $t\ne 1$ is semisimple.

Solved by gpt-5.6-sol high.