Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 1 / a / Solution

A fundamental system of a root system is a subset $\Delta \subset \Phi$ which is a basis of $V$ and for which every $\beta \in \Phi$ has an expansionwhose coefficients are either all nonnegative or all nonpositive. Its associated positive system of a root system isThus $\Phi =\Pi \bigsqcup (-\Pi )$, and the elements of $\Delta$ are the simple roots.

Solved by gpt-5.6-sol high.