Degree of a field extension

[Channel4]

Homework Statement

Let $ζ=e^((2*\pi*i)/7), E=Q(ζ), \xi=ζ + ζ^6$.

Show that $[Q(ζ):Q(\xi)]=2$.
Find the generator of the galois group $Gal(Q(ζ):Q(\xi))$.
What is the minimal polynomial of $\xi$.

Homework Equations

The Attempt at a Solution

I know that $[Q(ζ):Q]=6$ and that $Gal(Q(ζ):Q)$ is the cyclic group of order six. So I need to show that $[Q(\xi):Q]=3$, since $[Q(ζ):Q]=[Q(ζ):Q(\xi)][Q(\xi):Q]=6=2*3$.
Using $ζ^6+ζ^5+ζ^4+ζ^3+ζ^2+ζ+1=0$ and dividing by $ζ^3$ we get $ζ^3+ζ^2+ζ+1+ζ^-1+ζ^-2+ζ^-3=0=(ζ+ζ^-1)^3+(ζ+ζ^-1)^2-2(ζ+ζ^-1)-1=\xi^3+\xi^2-2\xi-1$. Not sure how to proceed from here. Any help would be greatly appreciated :).edit: Whats wrong with my latex?

 

[mighty2000]

There are a few errors in the latex formatting of your post. Here is a corrected version:

Homework Statement: Let $ \zeta = e^{\frac{2\pi i}{7}}, \ E = \mathbb{Q}(\zeta), \ \xi = \zeta + \zeta^6$.

Show that $[\mathbb{Q}(\zeta):\mathbb{Q}(\xi)] = 2$. Find the generator of the Galois group $\mathrm{Gal}(\mathbb{Q}(\zeta):\mathbb{Q}(\xi))$. What is the minimal polynomial of $\xi$?

Homework Equations:

The Attempt at a Solution: I know that $[\mathbb{Q}(\zeta):\mathbb{Q}] = 6$ and that $\mathrm{Gal}(\mathbb{Q}(\zeta):\mathbb{Q})$ is the cyclic group of order six. So I need to show that $[\mathbb{Q}(\xi):\mathbb{Q}] = 3$, since $[\mathbb{Q}(\zeta):\mathbb{Q}] = [\mathbb{Q}(\zeta):\mathbb{Q}(\xi)][\mathbb{Q}(\xi):\mathbb{Q}] = 6 = 2 \cdot 3$. Using $\zeta^6+\zeta^5+\zeta^4+\zeta^3+\zeta^2+\zeta+1=0$ and dividing by $\zeta^3$ we get $\zeta^3+\zeta^2+\zeta+1+\zeta^{-1}+\zeta^{-2}+\zeta^{-3}=0=(\zeta+\zeta^{-1})^3+(\zeta+\zeta^{-1})^2-2(\zeta+\zeta^{-1})-1=\xi^3+\xi^2-2\xi-1$. Not sure how to proceed from here. Any help would be greatly appreciated :).

The latex error in your post was mainly in the use of brackets and parentheses. In latex, brackets are denoted by "[" and "]" while parentheses are denoted by "\left(" and "\right)". Also, when using subscripts, it is better to use curly braces "{ }" instead of parentheses "()". Additionally, you can