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 indeed irreducible.

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 $\Q \sqbrk x$.

This needs considerable tedious hard slog to complete it. In particular: Factorise $\map P x$ by solving $\map P x = 0$ using Cardano's Formula for example, and note that the roots are irrational or complex or whatever. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$\blacksquare$

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