Is the cube root of a prime number rational? The question is: if $P$ is prime, is $P^{1/3}$ rational? 
I have been able to prove that if $P$ is prime then the square root of $P$ isn't rational (by contradiction) how would I go about the cube root? 

The main point is: The cube root of a natural number is rational iff it is infact an integer. More generally, any rational root of a monic  polynomial with integer coefficients (such as $X^3-n$) is in fact integer. So if $\sqrt[3] n$ is rational then $n$ is a cube (and cannot be prime).
