Let $a$ and $n$ be integers, such that $a,n>1$ and $n$ is not a perfect square; show that: $a^{\sqrt{n}}$ is a transcendental number.

Question

Although it is very hard to determine if a number is transcendental, I could appreciate any basic or simple insight or opinion concerning the statement, whether it is true or false.

Regards

Answer 1

It is true but not by any known "simple" insight but by a deep result known as the Gelfond theorem.The case a=n=2 was mentioned in David Hilbert's famous "Problem Set" over a century ago .The best I can do is to suggest some reading. If anyone wants to type out a complete proof of the Gelfond theorem here they'll probably get flagged.THEOREM (Gelfond): If $x,y$ are complex algebraic numbers , $0\ne x \ne1$, and $y$ not a rational real, then any value of $x^y$ is transcendental over the rationals...... For any integer $n$ let $l_n=i(2 \pi n+\arg x)+\ln |x|$ . We have $\exp (l_n)=x$ and we let $\exp (y l_n)$ be "a value of $x^y$".