# (Relatively) Trivial question about extension field degrees (pertains to Galois theor

1. May 7, 2012

### Elwin.Martin

So if we have an extension of E of F, then we can consider E as a vector space over F.
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 $\omega = cos (2\pi /7) + i sin (2\pi / 7)$

Why is it that:
"Since $x^7 -1 = (x-1)(x^6+x^5+x^4+x^3+x^2+x+1)$,
|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+cω2+...gω6 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]

2. May 7, 2012

### DonAntonio

Re: (Relatively) Trivial question about extension field degrees (pertains to Galois t

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