Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 20G / b / iii / Solution

The polynomial $T^3+3T+3$ is Eisenstein at $3$, so part (ii) givesFor a cubic $T^3+aT+b$, the polynomial discriminant is $-4a^3-27b^2$. HereThe discriminant-index formula for an integral lattice saysAny prime dividing the index must therefore occur to at least the second power in $351$; the only possibility is $3$, which has already been excluded. The index is one, and hence