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### Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers. [duplicate]

Take $p_1, p_2, \ldots, p_n, p_{n+1}$ be $n+1$ prime numbers in $\mathbb{P} \subseteq \mathbb{N}$. $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ seems to be quite ...
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### Exercise about field extensions [duplicate]

Consider $a_1,\ldots,a_n\in \mathbb Z$. i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$ Once ...
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### Dimensions of Field Extensions [duplicate]

I have a Galois theory exercise to prove that $$[\mathbb{Q}(\sqrt2,\sqrt3,\sqrt5):\mathbb Q]=8$$ I understand the proof up until the bit where If I prove $$\sqrt5 \notin \mathbb{Q}(\sqrt2,\sqrt3)$$ ...
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### An infinite extension of $\mathbb Q$ [duplicate]

Let $S=\{\sqrt p \in \mathbb R | p$ is a primer number$\}$. How can I show that $\mathbb Q(S)|\mathbb Q$ is an infinite field extension?