Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 7E / d / i / Solution

This can be false. For any nonzero transcendental number $\beta$, the number $\gamma =1/\beta$ is also transcendental, since otherwise its reciprocal would be algebraic. Yet $\beta \gamma =1$ is algebraic.

Solved by gpt-5.6-sol high.