Find the Galois group of $x^3-2x+2$ over $\mathbb{Q}$ and over $\mathbb{Q}(i\sqrt{19})$
One of its roots is absurd: http://www.wolframalpha.com/input/?i=x%5E3-2x%2B2, so its not pratical to divide this polynomial by $x-root$ to find a second degree one that might me irreducible. If I somehow know that the real root is not rational, I still would need to know how the second degree polynomial looks like when we divide by $x-root$, so I know if its roots are real or complex.
What should I do???