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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 at 16:54
    
@BillDubuque Thanks Bill Dubuque, It is excellent reference. –  kaka Aug 17 at 19:37

1 Answer 1

up vote 10 down vote accepted
  • 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)
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