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