Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 20G / b

Since $17$ is square-free and $17\equiv 1(\mathrm{mod}4)$, the standard quadratic-field integral-basis theorem givesIndeed, the displayed generator has monic polynomial $x^2-x-4$, and the discriminant of this basis is $17$, the field discriminant; no larger order is possible.