Galois Theory - irreducibility over Q

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To show that xn - a is irreducible over Q for a > 1 as a product of distinct primes and for all n ≥ 2, the rational root theorem can be considered, although it only confirms the absence of linear factors. Eisenstein's criterion is also applicable and can be a useful tool for establishing irreducibility in this case. The initial confusion stemmed from the belief that these methods might not be relevant, but further discussion clarified their usefulness. Overall, the problem can be approached effectively using these established criteria. Understanding these tools simplifies the process of proving irreducibility in polynomial equations.
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Homework Statement



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

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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.
 
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Hi Kate2010! :smile:

How about the rational root theorem?
 
I like Serena said:
Hi Kate2010! :smile:

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.
 
Dick said:
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. :)
 
Thanks guys - I was trying to make things more complicated than they were.
 

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