Let us look at factorization over the integers of polynomials of the form $x^n+n$. For the first few values of $n$ we get
- $x+1$ - irreducible
- $x^2+2$ - irreducible
- $x^3+3$ - irreducible
- $x^4+4$ - irred- oh, wait, this one can be factored as $(x^2-2x+2)(x^2+2x+2)$
- $x^5+5$ - irreducible
I tested for up to $n=20$ and all of them except for $x^4+4$ are irreducible.
So is there any other $n$ such that the polynomial $x^n+n$ can be factorised? If not, then why did $x^4+4$ break the pattern?
These are special cases of the following
Theorem $\ $ Suppose $\,F\,$ is a field and $\:a\in F\:$ and $\:0 < n\in\mathbb Z.\ $ Then
$\ \ \ x^n\! - a\, $ is irreducible over $\,F \iff a \not\in F^{\large p}\:$ for all primes $\:p\mid n,\:$ and $\ a\not\in -4\,F^4\,$ if $\: 4\mid n $
A proof can be found in many textbooks, e.g. Karpilovsky, Topics in Field Theory, Theorem 8.1.6 or Lang's Algebra (section on Galois Theory).