 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ia/paper-4/2e/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

r = p / q

p, q

and

q > 0

. Substituting into the monic polynomial and multiplying by

q^{n}

gives

p^{n} + a_{n - 1} p^{n - 1} q + \dots + a_{1} p q^{n - 1} + a_{0} q^{n} = 0.

All terms except the first are divisible by

q

, so

q ∣ p^{n}

. If

q > 1

p^{n}

, and the Euclid lemma implies that this prime divides

p

, contradicting coprimality. Thus

q = 1

r \in Q ⟹ r \in Z .

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