Schoenemann's Theorem

If the integral Coefficients $C_0$, $C_1$, ..., $C_{N-1}$ of the Polynomial

$$f(x)=C_0+C_1x+C_2x^2+\ldots+C_{N-1}x^{N-1}+x^N$$

are divisible by a Prime Number $p$, while the free term $C_0$ is not divisible by $p^2$, then $f(x)$ is irreducible in the natural rationality domain.

See also Abel's Irreducibility Theorem, Abel's Lemma, Gauss's Polynomial Theorem, Kronecker's Polynomial Theorem

References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 118, 1965.