Find the galois group of the polynomial when a root is given

Question

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ over $\mathbb{Q}$.

I am just introduced to field theory. I can determine the Galois group of the splitting field of the given polynomial by computing its discriminant.

But i have no idea how can we find the Galois group by the use of given roots.

Any hints will be highly appreciated. Thank you.

Answer 1

Here's a very computational start...

To show that $2-2\alpha-\alpha^2$ is a root, we can just use brute force: $$\begin{eqnarray*}f(2-2\alpha-\alpha^2)&=&\left(-\alpha ^2-2 \alpha +2\right)^3+\left(-\alpha ^2-2 \alpha +2\right)^2-4 \left(-\alpha ^2-2 \alpha +2\right)+1\\ &=&-\alpha ^6-6 \alpha ^5-5 \alpha ^4+20 \alpha ^3+16 \alpha ^2-24 \alpha +5 \\ &=&-\left(\alpha ^3+\alpha ^2-4 \alpha +1\right) \left(\alpha ^3+5 \alpha ^2+4 \alpha -5\right)\\ &=&-0\times \left(\alpha ^3+5 \alpha ^2+4 \alpha -5\right)\\ &=&0 \end{eqnarray*}$$ Now that that's verified, we know that $\alpha \mapsto 2-2\alpha-\alpha^2$ is an element of the Galois group. So let's plug $2-2\alpha-\alpha^2$ into $\alpha \mapsto 2-2\alpha-\alpha^2$ to get $$-\alpha ^4-4 \alpha ^3+2 \alpha ^2+12 \alpha -6\equiv \alpha ^2+\alpha -3\pmod{\alpha ^3+\alpha ^2-4 \alpha +1}$$ And of course then plugging $\alpha ^2+\alpha -3$ into $\alpha\mapsto -\alpha ^2-2 \alpha +2$ yields $$-\alpha ^4-2 \alpha ^3+3 \alpha ^2+4 \alpha -1\equiv \alpha \pmod{\alpha ^3+\alpha ^2-4 \alpha +1}$$ That means $\alpha \mapsto 2-2\alpha-\alpha^2$ is a three cycle.

Answer 2

I think that if you note that the sum of the roots of $f(x)$ is $1$ then you can see that all the roots of $f$ are linear combinations of $\alpha$. So all the roots of $f$ are contained in $ \mathbb{Q}(\alpha)$. Any splitting field contains $ \mathbb{Q}$ and $\alpha$ so this is the splitting field of $f$ over $\mathbb{Q}$.

What is an element of the Galois group then? How do elements of the Galois group act on the roots of the polynomial? What might be some possibilities and relations between them?