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I have received these problems and I'm not sure where to start:

Are these polynomials irreducible in $\Bbb Z[X]$ ?

a) $X^{2^n} + 1$ where $n$ is a positive integer.

b) $X^{p-1} + X^{p-2} + \cdots + X + 1 $ where $p$ is prime.

I know how to apply Eisenstein's Criterion, Gauss' Lemma and the Reduction Criterion but none of these seem to apply.

If anyone could explain any of these to me or give me a clue as to how to get started I'd be grateful.

Community
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stefan-niculae
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1 Answers1

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These can all be solved with the same method: to show $f(x)$ irreducible, apply Eisenstein to $f(x+1)$.

Andrew Dudzik
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  • Thank you for your answer but can you help me just a little bit more?

    For point (a), if I try to apply Eisenstein then I need a prime number p that will divide all the quotients except the last one. If I substitute for f(x+1), the quotients of the first few terms will be: 1, 65536, 6553565536 /2, 6553465535*65536 /6,and so on. What can that prime number be?

     – stefan-niculae Mar 16 '14 at 17:58
  • @user48345 The only one that divides 65536, of course. Or, there is always the constant term 1+1. :) – Andrew Dudzik Mar 16 '14 at 18:19