Question

Let H be a subgroup $\left\{i,\alpha \right\}$of ${\mathbf{Gal}}_{\mathbf{Q}}\mathbf{Q}\left(\sqrt{3},\sqrt{5}\right)\mathbf{=}\left\{i,\tau ,\alpha ,\beta \right\}$. Show that the fixed field of H is $\mathbf{Q}\left(\sqrt{3}\right)$. [Hint: Verify that $\mathbf{Q}\left(\sqrt{3}\right)\mathbf{\subseteq }{\mathbf{E}}_{\mathbf{H}}\mathbf{\subseteq }\mathbf{Q}\left(\sqrt{3},\sqrt{5}\right)$; what is $\mathbf{Q}\left(\sqrt{3},\sqrt{5}\right)\mathbf{:}\mathbf{Q}\left(\sqrt{3}\right)$?]

Short answer

$\mathrm{Q}\left(\sqrt{3}\right)\subseteq {\mathrm{E}}_{\mathrm{H}}\subseteq \mathrm{Q}\left(\sqrt{3},\sqrt{5}\right)$

Step-by-step solution

Automorphism:

The Galols group is denoted by Gal_FK when the set of all F automorphism of K is an extension of F.

It is the Galols group of K over F.

To find Q(3,5):Q(3)   

Let H contains the elements $\left(i,\alpha \right)$of the Galols group of ${\mathrm{Gal}}_{\mathrm{Q}}\mathrm{Q}\left(\sqrt{3},\sqrt{5}\right)$

And also H be the subgroup of fixed field E.

The Galols group of $\mathrm{Q}\left(\sqrt{3},\sqrt{5}\right)$ is considered over Q.

The two minimal polynomial equations are ${\mathrm{x}}^2-3$and ${\mathrm{x}}^2-5$.

Which takes a root $\sqrt{3}$to $\sqrt{3}$or $-\sqrt{3}$and similarly for other polynomials.

There are atmost 4 automorphism in given $Ga{l}_{Q}Q\left(\sqrt{3},\sqrt{5}\right)$over Q such that there are two possibilities:

${\sqrt{3}}^{\mathrm{i}}\to \sqrt{3},{\sqrt{3}}^{\mathrm{r}}\to \sqrt{3},{\sqrt{3}}^{\mathrm{\alpha }}\to \sqrt{3},{\sqrt{3}}^{\mathrm{\beta }}\to -\sqrt{3}$

Same as,

${\sqrt{5}}^{\mathrm{i}}\to \sqrt{5},{\sqrt{5}}^{\mathrm{r}}\to \sqrt{5},{\sqrt{5}}^{\mathrm{\alpha }}\to \sqrt{5},{\sqrt{5}}^{\mathrm{\beta }}\to -\sqrt{5}$

Now, the minimal polynomial is isomorphism.

Now, H $\left(i,\alpha \right)$

So, $\mathrm{i}\sqrt{3}=\sqrt{3}$

And, $\alpha \sqrt{3}=\sqrt{3}$

Now, find $Ga{l}_{Q}Q\left(\sqrt{3},\sqrt{5}\right)$over the field $Q\left(\sqrt{3}\right)$.

Therefore, $\mathrm{H}\in \mathrm{E}$ and

$E_H=Q\left(\sqrt{3}\right)$

Thus, the range to decide Galol group is $\mathrm{Q}\left(\sqrt{3}\right)\subseteq {\mathrm{E}}_{\mathrm{H}}\subseteq \mathrm{Q}\left(\sqrt{3},\sqrt{5}\right)$.