Schönemann-Eisenstein Theorem/Warning

Schönemann-Eisenstein Theorem: Warning

The converse of the Schönemann-Eisenstein Theorem does not hold.

That is, if a polynomial over $\Z$ is irreducible in $\Q \sqbrk x$, it is not necessarily the case that the criteria:

$(1): \quad p \divides a_i \iff i \ne d$

$(2): \quad p^2 \nmid a_0$

where:

$p$ is a prime

$p \divides a_i$ signifies that $p$ is a divisor of $a_i$

both hold.

See Schönemann-Eisenstein Theorem: $x^3 + 2 x + 4$ for a counterexample.