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

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

score 16 · Accepted Answer · answered Aug 17 '14 at 06:55

16

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)

answered Aug 17 '14 at 06:55

1 Answers1