I am being asked this question as an exercise in Garling's "A course in Galois theory". But isn't this an open question in math?
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Asaf Karagila
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Temitope.A
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3Like $e^{i\pi}$? – anon Aug 21 '13 at 14:45
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8No: $e$ and $\log2$ are transcendental, but $2$ is not. – Andrés E. Caicedo Aug 21 '13 at 14:45
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@Andres: That should be an answer. – Asaf Karagila Aug 21 '13 at 14:46
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@AsafKaragila Go ahead. – Andrés E. Caicedo Aug 21 '13 at 14:47
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Yeah. That's indeed an easy counterexample. Are there others which are not of the form $re^{iaπ}$. With others I mean simple ones. – Temitope.A Aug 21 '13 at 14:51
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No. We know $e$ and $\log(2)$ are transcendental, but $$e^{\log 2}=2.$$
Doug Chatham
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Cameron Buie
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1I know what you mean. I had a similar reaction when I learned that every value of $i^i$ is real. – Cameron Buie Aug 21 '13 at 15:07