# Degree of a field extension

1. Apr 27, 2013

### Channel4

1. The problem statement, all variables and given/known data
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 $\xi$.

2. Relevant equations

3. 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?

Last edited: Apr 27, 2013