Schönemann-Eisenstein Theorem/Examples/x^3 - 2x + 4

Example of Use of Schönemann-Eisenstein Theorem

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 $\Q \sqbrk x$.

Proof

We note that the prime number $2$:

is a divisor of the coefficient of $x^1$, that is, $-2$

is not a divisor of the degree of $\map P x$, that is, $3$

and that:

$2^2$ is a divisor of the coefficient of $x_0$, that is, $4$.

Hence, by the Schönemann-Eisenstein Theorem, it is not necessarily the case that $\map P x$ is not irreducible over $\Q \sqbrk x$.

However, we note that:

$x^3 - 2 x + 4 = \paren {x + 2} \paren {x^2 - 2 x + 2}$

demonstrating that $\map P x$ is indeed not irreducible over $\Q \sqbrk x$.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Eisenstein's criterion
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Eisenstein's criterion