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1)

Let K be the algebraic closure of a field F and suppose E is a field such that F [tex]F \subseteq E \subseteq K[/tex]. Then is K the algebraic closure of E?

2)

Let [tex]n[/tex] be a natural number with [tex]n\geq2[/tex], and suppose that [tex]\omega[/tex] is a complex nth root on unity. Is there a formula for [tex]\left[\mathbb{Q}(\omega) : \mathbb{Q}\right][/tex] ?

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To 1), I must be missing something really silly, because it seems to me like it is obviously the case that K is also the algebraic closure of E, and that the proof should be easy. But I simply can't think of anything.

To 2) I would say no, but I am not exactly sure that I understand the question. For example, if n=8, then [tex]e^{i2\pi/8}[/tex] is a complex 8th root of unity such that [tex]\left[\mathbb{Q}(e^{i2\pi/8}) : \mathbb{Q}\right]=4[/tex]. However, [tex]i[/tex] is also an 8th root of unity, but [tex]\left[\mathbb{Q}(i) : \mathbb{Q}\right]=2[/tex]. Thus, for a given n, there is not necessarily a formula. Does this sound right?