2
$\begingroup$

I have a question concerning definition in terms of minimal polynomial i.e. if we let $E = F(\alpha)$ be a field extension of $F$ of degree two then how do I describe, in terms of the minimal polynomial for $\alpha$ over $F$ when this field extension is Galois?

Also does there exist a field extension of $\mathbb{Q}$ of degree $3$ that is Galois?

$\endgroup$
3
$\begingroup$

In general, $F(\alpha)/F$ is Galois if and only if the minimal polynomial of $\alpha$ over $F$ is separable and splits completely in $F(\alpha)$. In the special case of quadratic extensions, it is very easy to see that the "splits completely" part is always satisfied, so it's just a question of separability. If for example $F$ has characteristic 0, then separability is automatic, so in that case, any quadratic extension is Galois.

There do exist Galois extensions of $\mathbb{Q}$ of degree 3. If $\alpha$ has cubic minimal polynomial over $\mathbb{Q}$, then $\mathbb{Q}(\alpha)$ is Galois if and only if the discriminant of this polynomial is a rational square. This is proved in any standard text on Galois theory.

$\endgroup$
1
$\begingroup$

Let $\alpha=e^{2\pi i/7}$, and let $\beta=\alpha+\alpha^{-1}$. You should be able to find the conjugates of $\beta$, prove that they are all in ${\bf Q}(\beta)$, and that that field is Galois of degree 3 over the rationals.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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