Galois Theory - irreducibility over Q

[Kate2010]

Homework Statement

If a>1 is a product of distinct primes, show that xⁿ-a is irreducible over Q for all n ≥ 2.

Homework Equations

The Attempt at a Solution

I am not really sure how to start this problem. Can anyone point me in the right direction?

I know tests for irreducibility for example Eisensteins Criterion or reduction modulo p but I don't think that these are helpful here?

Thanks for any help.

 

[I like Serena]

Hi Kate2010!

How about the rational root theorem?

 

[Dick]

Hi Kate2010!

How about the rational root theorem?

That would only tell you it doesn't have any linear factors. I'm a little confused why Kate2010 thinks Eisenstein's criterion isn't applicable.

 

[I like Serena]

That would only tell you it doesn't have any linear factors. I'm a little confused why Kate2010 thinks Eisenstein's criterion isn't applicable.

Right.
Just looked up Eisenstein's criterion.
Looks like a good one. :)

 

[Kate2010]

Thanks guys - I was trying to make things more complicated than they were.