Irreducibility of $x^3-6x-2$ in $Q[x]$

Question

Please help me to answer the following problem:

Let $f(x)=x^3-6x-2\in\mathbb{Q}[x]$ and $L$ be a splitting field of $f(x)$ over $\mathbb{Q}$. Show that $f$ is irreducible in $\mathbb{Q}[x]$ and compute the discriminant of $f(x)$.

Thanks

Answer 1

Use the Eisenstein's criterion for $p=2$.

See here: https://en.wikipedia.org/wiki/Eisenstein%27s_criterion

Assume that $x_i$ they are roots of $f$, $x_1+x_2+x_3=3u$, $x_1x_2+x_1x_3+x_2x_3=3v^2$ and $x_1x_2x_3=w^3$.

Hence, $u=0$, $v^2=-2$ and $w^3=2$.

Thus, $$\Delta=(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2=27(3u^2v^4-4v^6-4u^3w^3+6uv^2w^3-w^6)=756$$

Answer 2

You can use Rational root theorem and observe that $\pm1,\pm2$ are not roots.

Indeed each rational solution x must be in the form $x = \frac{p}{q}$ with

• p integer factor of the constant term $a_0=-2$
• q integer factor of the leading coefficient $a_3=1$.

Answer 3

you can use irreducible mod $p$. If $p=5$, then $x^3-6x-2\equiv x^3+4x+3\pmod 5$ and we note that $x^3+4x+3$ is irreducible in $\Bbb Z_5[x]$

Discriminant of $f(x)$ is $-(4(-6)^3+27(-2)^2)=756$.