(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the splitting field of x^{8}− 2 over Q, including its degree

2. Relevant equations

Degree is multiplicative in tower of fields:

F [tex]\subset[/tex] K [tex]\subset[/tex] 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([tex]\sqrt[8]{2}[/tex], [tex]\zeta[/tex]).

Q[tex]\subset[/tex]Q([tex]\sqrt[8]{2}[/tex]) [tex]\subset[/tex] K

[K: Q([tex]\sqrt[8]{2}[/tex])] = 8

=> 8 | [K : Q]

Q[tex]\subset[/tex]Q([tex]\zeta[/tex]) [tex]\subset[/tex] K

[K: Q([tex]\sqrt[8]{2}[/tex])] = 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)

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# Homework Help: Degree of Field Extension

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