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The dimension of this space is the degree of the field extension, I think most people use [E:F].

This is correct in most people's books, right?

Defining [itex] \omega = cos (2\pi /7) + i sin (2\pi / 7) [/itex]

Why is it that:

"Since [itex] x^7 -1 = (x-1)(x^6+x^5+x^4+x^3+x^2+x+1) [/itex],

|Gal(Q(ω)/Q)|=[Q(ω):Q]≤ 6"

?

With another piece of information he gets that the degree is 6. Since the vector space of Q(ω) over Q has degree 6, we should be able to represent it with 6 basis vectors...but this just feels weird since ω is a seventh root of unity and it feels like we should have something like:

a+bω+cω

^{2}+...gω

^{6}a,b,...,g in Q, right?

What am I missing? I have no reason to doubt Gallian, I even looked up the general method for calculating degrees of Q(n-th roots of unity)/Q and I know that 6 is correct. . .I just don't see why and I can't construct 6 basis vectors.

[This is from Gallian's Abstract Algebra book, 7th edition, in Ch. 32 on page 550]