Irreducibility of Polynomial in $\mathbb{Q}$

Question

How can we show that, if $a>1$ is the product of distinct primes, then $x^n-a$ is irreducible in $\mathbb{Q}$ for all $n \geq 2$ and that it has no repeated roots in any extension of $\mathbb{Q}$?

Thoughts:

Perhaps we can do an inductive proof on $x^{k+2}-a$ as the $k=0$ and $k=1$ cases I know how to deal with.

Any help would be appreciated.

Answer 1

Hint: Eisenstein's Criterion.

Answer 2

For irreducibility use Eisenstein's cirterion (with any of the primes dividing $a$).

A repeated root of $f(x)$ would be a common root of $f(x)$ and $f'(x)=nx^{n-1}$, hence also a root of $nf(x)-xf'(x)=-na$

(Note that both parts work for $n\ge 1$).