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## Homework Statement

Let S= {e^2*i*pi/n for all n in the natural numbers} and let F=Q

Is F:Q

1) algebraic?

2) finite?

3) simple?

4)separable?

## Homework Equations

## The Attempt at a Solution

1) Every element in S is a root of x^n-1 and every element of a in Q is a root of x-a, and thus I think somehow that means that the whole extension is algebraic is (i.e all the basis elements are algebraic and obviously all the elements of the base field are algebraic). Does this argument work? If not what would?

2) No. Every p in the natural numbers will be a basis that is linearly independent of all the elements before it, and there are infinite primes, so this will not be a finite extension.

3) I'm not sure. I want to say there was some theorem that algebraic extensions can ultimately be seen as simple but I'm not sure.

4) Yes, this extension is over Q where char(Q) = 0 so the extension is separable.

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