Computing the galois group of $x^3+3x+1$

Question

I want to calculate the galois group of the polynomial $x^3+3x+1$ over $\mathbb Q$.

And I am struggling in finding the roots of the polynomial.

I only need a tip to start with. Not the full solution of this task.

Thanks :)

Answer 1

How many real roots does it have? (Use calculus.)

Answer 2

Hint: The Galois group is a transitive subgroup of $S_3$ (courtesy of Transitive Group Data).

Combine this with the answer you get from hunter's hint.

Answer 3

One of the roots is

$$\alpha^{\frac{1}{3}}-\alpha^{-\frac{1}{3}}\quad \text{ where }\quad \alpha = \frac{\sqrt{5}-1}{2}.$$

In general, for $x^3 + 3px + 2q = 0$ we have the real root (there is surely at least one) given by

$$\alpha^{\frac{1}{3}}-p\alpha^{-\frac{1}{3}}\quad\text{ for }\quad\alpha = \sqrt{p^3+q^2}-q.$$

I hope this helps $\ddot\smile$