TITLE: Is product of algebraic and transcendental number transcendental?
QUESTION [6 upvotes]: Let $\alpha \in \mathbb{A}$, and  $\gamma \in \mathbb{T}$.
 I know  that the reciprocal of a transcendental number is transcendental.
Question:

Is  $\alpha\cdot \gamma \in \mathbb{T}$?

REPLY [14 votes]: No if $\alpha=0$ (because then you get $0$ which is algebraic)
Yes if $\alpha \neq 0$ (because if $\alpha \gamma$ is algebraic, then also $\alpha^{-1} \alpha \gamma = \gamma$ would be algebraic)