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I just want to know:

If a certain number is transcendental, call it $n$, is it safe to say that $n^2$ or that multiples of $n$ are are also transcendental?

For example, from $e$ is transcendental, can we deduce that $e^2$ is transcendental?

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user138798
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1 Answers1

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If $\alpha$ is transcendental, and $P(x)$ is a non-constant polynomial with algebraic coefficients, then $P(\alpha)$ is transcendental.

In particular, $e^2$ is transcendental (let $P(x)=x^2$).

André Nicolas
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  • where can you find a proof of this theorem? – Christian Chapman May 14 '14 at 20:32
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    Suppose that $P(\alpha)$ is algebraic. Then $P(\alpha)$ is a root of a polynomial $Q(x)$ with algebraic coefficients. But then $F(x)=Q(P(x))$ is a polynomial with algebraic coefficients, and $F(\alpha)=0$. This is impossible, since $\alpha$ is transcendental, so is not the root of any non-zero polynomial with algebraic coefficients. – André Nicolas May 14 '14 at 20:45
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    Good heavens, i should have seen that. Thank you! – Christian Chapman May 14 '14 at 20:47