Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
1  
So, why does this question have many upvotes and is on the hot network questions?! –  metacompactness May 14 at 21:38

1 Answer 1

up vote 7 down vote accepted

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 $P(x)=x^2$).

share|improve this answer
    
where can you find a proof of this theorem? –  enthdegree May 14 at 20:32
1  
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 at 20:45
1  
Good heavens, i should have seen that. Thank you! –  enthdegree May 14 at 20:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.