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I want to know how to calculate the degree of the field extension $[K:Q]$ where $K$ is the splitting field of $x^{13}+1$ over $\mathbb{Q}$.

I'm new to this area and this is not really covered in my course properly. So please don't assume I'm familiar to much when answering.

  1. Since $-1$ is a root should I conclude that all roots are $-1w^{n}$, where $w\in\mathbb{C}$ and $w^{13}=1$ or am I searching for the solutions to $x^{13}=-1 \in\mathbb{C}$, or is this just the same thing since $-1\in\mathbb{Q}$ already?

  2. How do I go about finding solutions to these equations in $\mathbb{C}$? After finding solutions how do I know which are the minimal polynomials satisfying these?

A lot of questions at the same time, but I don't really have anyone else to ask. Btw this is not a school assignment!

Are the roots $-w^{n}$ where $1\leq$n$\leq12$ and $w=\mathbb{e}^{\frac{2\pi}{13}i}$? And if we are searching for the $n$'th roots of unity when $n$ is composite why do we only include powers that are coprime to $n$?

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    Look up (complex) roots of unity, [cyclotomic polynomials](http://en.wikipedia.org/wiki/Cyclotomic_polynomial), and show that $K=\mathbb{Q}(e^{\pi i/13})$.2012-08-13
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    What do you mean by "we only include powers that are coprime to $n$"?2012-08-13

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