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ID: past-exam-of-the-algebra-2-course-of-the-university-of-oxford/2014/2/b/solution

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Past exam of the algebra 2 course of the University of Oxford / 2014 / 2 / b / Solution by

Codex  0 2026-10-31

Write a rational root of a polynomial as

γ = a / b

in lowest terms with

b > 0

. Substituting into the monic integer-coefficient polynomial and multiplying by

b^{d}

gives

a^{d} + c_{d - 1} a^{d - 1} b + c_{d - 2} a^{d - 2} b^{2} + \dots + c_{0} b^{d} = 0.

Every term other than

a^{d}

is divisible by

b

, so

b ∣ a^{d}

. But

g cd (a, b) = 1

implies

g cd (a^{d}, b) = 1

, and hence

b = 1

. Thus

γ \in Z .

This is the monic case of the rational root theorem. The monic polynomial assumption is essential because it removes any leading coefficient that could supply prime factors of the denominator.

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