Irreducible Polynomial/Examples/8 x^3 - 6 x - 1 in Rationals

Examples of Irreducible Polynomials
Consider the polynomial:
 * $\map P x = 8 x^3 - 6 x - 1$

over the ring of polynomials $\Q \sqbrk X$ over the rational numbers.

Then $\map P x$ is irreducible.

Proof
$\map P x$ has proper factors.

Then one of these has to be of degree $1$.

Thus from Factors of Polynomial with Integer Coefficients have Integer Coefficients we have:
 * $8 x^3 - 6 x - 1 = \paren {a x + b} \paren {c^2 + d x + e}$

for some $a, b, c, d, e \in \Z$.

Hence:

The only possible degree $1$ factors with integer coefficients are:


 * $x \pm 1, 2 x \pm 1, 4 x \pm 1, 8 x \pm 1$

By trying each of these possibilities, it is determined that no integer value of $d$ gives the correct values.

Hence the result.