Schönemann-Eisenstein Theorem/Examples

Examples of Use of Schönemann-Eisenstein Theorem

Example: $x^3 + 2 x + 2$

Consider the polynomial:

$\map P x = x^3 + 2 x + 2$

By the Schönemann-Eisenstein Theorem, $\map P x$ is irreducible over $\Q \sqbrk x$.

Example: $x^3 + 2 x + 4$

Consider the polynomial:

$\map P x = x^3 + 2 x + 4$

By the Schönemann-Eisenstein Theorem, it is not possible to tell whether or not $\map P x$ is irreducible over $\Q \sqbrk x$.

In fact, in this case $\map P x$ is indeed irreducible.

Example: $x^3 - 2 x + 4$

Consider the polynomial:

$\map P x = x^3 - 2 x + 4$

By the Schönemann-Eisenstein Theorem, it is not possible to tell whether or not $\map P x$ is irreducible over $\Q \sqbrk x$.

In fact, in this case $\map P x$ is not irreducible over $\Q \sqbrk x$.