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

Examples of Irreducible Polynomials

Consider the polynomial:

$\map P x = 8 x^3 - 6 x - 1$

Then $\map P x$ is irreducible.

Aiming for a contradiction, suppose $\map P x$ has proper factors.

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

$8 x^3 - 6 x - 1 = \paren {a x + b} \paren {c^2 + d x + e}$

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

Hence:

$\ds a c$

$=$

$\ds 8$

$\ds \leadsto \ \ $

$\ds a$

$\divides$

$\ds 8$

$\ds b e$

$=$

$\ds -1$

$\ds \leadsto \ \ $

$\ds b$

$\divides$

$\ds 1$

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.

$\blacksquare$

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 31$. Polynomials with Integer Coefficients: Example $59$