The smallest Galois extension of $\mathbb{Q}(x^3)$ containing $\mathbb{Q}(x)$

What's the smallest Galois extension of $\mathbb{Q}(x^3)$ containing $\mathbb{Q}(x)$?

For example, let E be the smallest Galois extension of $\mathbb{Q}(x^3)$ containing $\mathbb{Q}(x)$. Then, do i have to show that $E/\mathbb{Q}(x^3)$ is a separable extension and a normal extension?

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It depends on your definitions. If a Galois extension is defined to be a separable normal extension, then there's nothing to show. – Zhen Lin Dec 20 '11 at 13:36
Thank you for comment! – sunyeul Dec 20 '11 at 16:26

1 Answer

The minimal polynomial of $x$ over $\mathbb{Q}\left(x^3\right)$ is

$$f(X) = X^3 - x^3,$$

since it is irreducible (by the Eisenstein criterion, for example), and has clearly $x$ as a root.

The splitting field of $f(X)$ is $\mathbb{Q}(\omega, x)$, where $\omega$ is any primitive third root of unity.

So $\mathbb{Q}(\omega, x)$ is the smallest Galois extension of $\mathbb{Q}\left(x^3\right)$ containing $x$.

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