Schönemann-Eisenstein Theorem/Examples/x^3 - 2x + 4
Jump to navigation
Jump to search
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