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

Example of Use of Schönemann-Eisenstein Theorem

Consider the polynomial:

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

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

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 not a divisor of the coefficient of $x_0$, that is, $2$.

Hence, by the Schönemann-Eisenstein Theorem, $\map P x$ is 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