Irreducibility of $X^n-p^m$

Asked Nov 29 '16 at 21:37

Active Nov 29 '16 at 21:37

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I'm trying to show the following:

Let $p$ be a prime numer and $n,m \geq 1$ prime. $$f(X):=X^n-p^m$$ is irreducible in $\mathbb{Q}[X]$. Also, this statement is generally wrong, if $m, n$ aren't prime.

I tried to use the Eisenstein-criteria, but I don't think it works here (or at least it didn't work out, when I tried). For the last part I thought about simply showing a counter-example, but I can't think of a proper one...

Could someone help me, please?

asked Nov 29 '16 at 21:37

Tobi92sr

1,661

The claim is false if $n=m$. – N. S. Nov 29 '16 at 21:39
@N.S. But then $n$ and $m$ are not coprime. (I suppose the OP means "if $m,n$ aren't coprime"). – Dietrich Burde Nov 29 '16 at 21:40
@DietrichBurde The problem says prime, not co-prime. – N. S. Nov 29 '16 at 21:40
@N.S. I kinda overlooked that possibility... Thank you. – Tobi92sr Nov 29 '16 at 22:08
@DietrichBurde Hmm... I don't see the other problem being similar to mine. If the other post has anything to do with Galois-Theory, I'm afraid, I haven't learned anything about that yet. If there is any other connection between the two posts, would you mind explaining it to me? – Tobi92sr Nov 29 '16 at 22:19
You can apply Bill's answer for $c=p^m$, and also look into the book he has cited. For the counterexample you can use this MSE-question. And your sentence "Also, this statement is generally wrong, if m,n aren't prime" is not correct. – Dietrich Burde Nov 30 '16 at 09:14
@DietrichBurde I'm sorry, but I don't understand most of the notations in the book he recommends and I can't really follow the steps. Would you mind explaining it to me shortly? – Tobi92sr Nov 30 '16 at 23:49
This is actually more of an duplicate of this question: Irreducibility of $f(x)=x^n-p^m$ . – Sil Jun 01 '18 at 16:47

Irreducibility of $X^n-p^m$

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