Degree of Field Extension

[jk1100]

Homework Statement

Find the splitting field of x⁸ − 2 over Q, including its degree

Homework 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)|

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)

 

[mighty2000]

...Hello,

To find the splitting field of x^8 - 2 over Q, we first need to factor the polynomial into irreducible factors. We can use the fact that x^8 - 2 is a cyclotomic polynomial, as it can be written as (x^8 - 1) - (2 - 1). This means that it can be factored into the product of the cyclotomic polynomials of the 8th roots of unity and (x-1) (since 2 is not a root of unity). So we have:

x^8 - 2 = (x^8 - 1) - (2 - 1) = (x-1)(x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) - 1 = (x-1)(x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) - 1

Now, we can see that the polynomial x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 is irreducible over Q by using the Eisenstein criterion with p = 2. This means that the splitting field of x^8 - 2 over Q is the field K = Q(\sqrt[8]{2}, \zeta_8), where \zeta_8 is a primitive 8th root of unity.

To find the degree of this field extension, we can use the fact that the degree is multiplicative in a tower of fields. We have Q \subset Q(\sqrt[8]{2}) \subset K, and we know that [Q(\sqrt[8]{2}) : Q] = 8 and [K : Q(\sqrt[8]{2})] = 8 (since K is the splitting field of x^8 - 2 over Q(\sqrt[8]{2})). This means that [K : Q] = [K : Q(\sqrt[8]{2})][Q(\sqrt[8]{2}) : Q] = 8*8 = 64.

Therefore, the splitting field of x^8 - 2 over Q is K = Q(\sqrt[8]{2}, \zeta_8), and its degree is 64