Is product of algebraic and transcendental number transcendental?

Question

Let $\alpha \in \mathbb{A}$, and $\gamma \in \mathbb{T}$. I know that the reciprocal of a transcendental number is transcendental.

Question:

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

Answer 1

• 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)