Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
1  
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
add comment

1 Answer 1

up vote 2 down vote accepted

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 $x$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.