Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 21 / 2 / b

Use the two-prime bound on rational elliptic torsion with the counts six at three and eight at five from solution. For a prime $\ell \ne 3,5$, the $\ell$-primary torsion subgroup injects into both reduced groups by reduction of torsion points on an elliptic curve. Its order therefore divides the corresponding primary part of $\gcd (6,8)=2$. The three-primary subgroup injects into the group of order eight at five and is trivial; the five-primary subgroup injects into the group of order six at three and is trivial. Hence the entire rational torsion subgroup has order at most two.

The factorization $x^3+x^2+x+1=(x+1)(x^2+1)$ supplies $T=(-1,0)$, a nonidentity point of order two under the elliptic curve group law. ThereforeThe primary-component argument avoids assuming injectivity on residue-characteristic torsion.