This question was asked in a quaifying exam for which I am preparing.
Let F $\subseteq \mathbb{C}$be the splitting field of $x^{7}-2$ over $\mathbb{Q}$and $z=e^{2\pi i /7}$, aprimitive 7th root of unity.
Then Find $[F:\mathbb{Q(z)}]$=a and $[F:\mathbb{Q}({2}^{1/7})]=b$
I have studied Field Theory from Thomas Hungerford and couldn't try any problems due to lack of time.So, I am not able solve it.
splitting field is a field in which polynomial splits completely ie all its roots are linear and $({2}^{1/7})$ is one of the roots. But I am not able to use them in my question .
Can you please help on how to tackle the problem. I know this is happening due to lack of problem solving.
Thanks!!
It is $|\mathbb{Q}(\sqrt[7]{2}):\mathbb{Q}|=7$ and $|\mathbb{Q}(z):\mathbb{Q}|=\phi(7)=6$. Hence $$|\mathbb{Q}(\sqrt[7]{2},z):\mathbb{Q}|=|\mathbb{Q}(\sqrt[7]{2},z):\mathbb{Q}(z)||\mathbb{Q}(z):\mathbb{Q}|=|\mathbb{Q}(\sqrt[7]{2},z):\mathbb{Q}(\sqrt[7]{2})||\mathbb{Q}(\sqrt[7]{2}):\mathbb{Q}|\Rightarrow$$
$$\Rightarrow |\mathbb{Q}(\sqrt[7]{2},z):\mathbb{Q}(\sqrt[7]{2})|\cdot 6=|\mathbb{Q}(\sqrt[7]{2},z):\mathbb{Q}(z)|\cdot 7\Rightarrow $$
$|\mathbb{Q}(\sqrt[7]{2},z):\mathbb{Q}(\sqrt[7]{2})|=7$ and $|\mathbb{Q}(\sqrt[7]{2},z):\mathbb{Q}(z)|=6$