Is $x^n - y^m$ irreducible ($n, m$ strictly positive integers with $n \ne m$)?

Asked Sep 12 '20 at 12:23

Active Sep 12 '20 at 13:26

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For $n$ and $m$ strictly positive integers with $n \ne m$, I guess that $x^m - y^n$ is irreducible in any field, but I am not sure. In particular, I have no idea how this could be proved. My question is thus:

Is $x^m - y^m$ irreducible ? In any field, or only in fields with some restriction?
If yes, how to prove it ? If no, is there a concrete counterexample?

edited Sep 12 '20 at 13:26

Davood

4,265

asked Sep 12 '20 at 12:23

Loic

3

$x^2-y^4=(x-y^2)(x+y^2)$. I think you need $\gcd(m,n)=1$. – Angina Seng Sep 12 '20 at 12:26
3

Does this answer your question? Necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$. Also $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$ (and linked questions) – Sil Sep 12 '20 at 12:29
Title: "irreductible" should be "irreducible". – Dietrich Burde Sep 12 '20 at 13:14

Is $x^n - y^m$ irreducible ($n, m$ strictly positive integers with $n \ne m$)?

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