Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 4 / b / iv / Solution

If ${\eta }_2$ is monic, then ${\eta }_2(0)\ne {\eta }_2(1)$. The free algebra $(T2,{\mu }_2)$ therefore has more than one element, proving the existence of a nontrivial $T$-algebra.