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I'm dealing with the following question:

$x^3 - 3x +1$ and $a,b,c$ are the roots of the polynomial. Verify that f splits $\mathbb{Q}(a)$. Express $b$ and $c$ in the basis ${1, a, a^2}$

My work: I found out that the discriminant was $81$ and hence a perfect square. All the roots are unequal and irrational. But, how can I show that the polynomial will split if I simply adjoin one of the roots to $\mathbb{Q}$? Is there any theorem regarding this issue that I'm missing? How do I express the other roots in terms of $a$? Will Cardano's formula help? When I try to use Cardano's formula, I get a very messy expression and I'm not sure how to use that to solve this problem.

ludwiggodel432
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  • Hint: Show that if $a$ is a root, then $a^2-2$ is another. This is a standard exercise. It has been discussed on this site earlier. Alternatively you can use the fact that a square discriminant forces the Galois group to be a transitive subgroup of $A_3$, so.... – Jyrki Lahtonen Oct 06 '21 at 16:05
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    See here. I think this is a duplicate. But as I happened to answer the target it is bad form to use my immediately binding dupehammer vote. – Jyrki Lahtonen Oct 06 '21 at 16:09
  • See https://math.stackexchange.com/questions/1767252/expressing-the-roots-of-a-cubic-as-polynomials-in-one-root – lhf Oct 06 '21 at 16:37

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