# Is product of algebraic and transcendental number transcendental?

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

Question:

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

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This works for many "number systems". See this answer, for a general viewpoint. – Bill Dubuque Aug 17 '14 at 16:54
@BillDubuque Thanks Bill Dubuque, It is excellent reference. – kaka Aug 17 '14 at 19:37

• 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)