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Which of the following are true?

1) Given any positive integer $n$, there exists a field extension of $\mathbb{Q}$ of degree $n$.

2) Given any positive integer $n$, there exist fields $F$ and $K$ such that $F\subseteq K$ and $K$ is Galois over $F$ with $[K:F]=n$.

3) Let $K$ be Galois extension of $\mathbb{Q}$ with $[K:\mathbb{Q}]=4$. Then there is a field $L$ such that $\mathbb{Q} \subseteq L \subseteq K$, $[L:\mathbb{Q}]=2$ and $L$ is a Galois extension of $\mathbb{Q}$.

4) There is an algebraic extension $K$ of $\mathbb{Q}$ such that $[K:\mathbb{Q}]$ is not finite

1) is true, because $\mathbb{Q} (2^{1/n})$ over $\mathbb{Q}$ is an example.

4) is also true ,as $[\mathbb{Q} (2^{1/2},2^{1/3},2^{1/4},...,) : \mathbb{Q}]$ is not finite.

How to approach 2) and 3) ?

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For 2) You can take $\mathbb F_p \subset \mathbb F_{p^n}$.

For 3) note that any group with four elements is abelian and certainly has a subgroup of order $2$. The galois correspondence now shows the existence of $L$.

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  • $\begingroup$ Now.....I'm clear! Thank a lot! $\endgroup$ – user444830 May 20 '17 at 10:12

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