# Homework Help: Degree of Field Extension

1. May 5, 2010

### jk1100

1. The problem statement, all variables and given/known data
Find the splitting field of x8 − 2 over Q, including its degree

2. Relevant equations
Degree is multiplicative in tower of fields:
F $$\subset$$ K $$\subset$$ L
[L:F] = [L:K][K:F]

Degree of Galois extension is equal to the order of the Galois group
[K:F] = |Gal(K/F)|

3. The attempt at a solution
The splitting field is gotten by adjoining the eighth root of 2 the a primitive eighth root of unity K=Q($$\sqrt[8]{2}$$, $$\zeta$$).

Q$$\subset$$Q($$\sqrt[8]{2}$$) $$\subset$$ K
[K: Q($$\sqrt[8]{2}$$)] = 8
=> 8 | [K : Q]

Q$$\subset$$Q($$\zeta$$) $$\subset$$ K
[K: Q($$\sqrt[8]{2}$$)] = 7
=> 7 | [K : Q]

So I believe 56 must divide the order of the field extension. I also know since K is the splitting field of the polynomial it in a Galois extension; so the degree must be the order of the Galois group. However I have tried, but I have had much difficultly getting a handle of the Galois group Gal(K/Q)