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Write $\epsilon = \epsilon_p$, where $p$ $>$ 2 is prime, and let $\alpha$ = $\sum_{i=0}^{p-1} \epsilon^{i^{2}}$. Show that $\mathbb {Q}$$[\alpha]$ is the unique subfield of $\mathbb {Q}_p$ that has degree 2 over $\mathbb {Q}$. I am having trouble wrapping my head around this problem. I have no idea how to start it.

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    Do you want to write $\mathbb Q(\epsilon)$ (I assume that $\epsilon$ is a primitive $p$-th root of $1$) instead of $\mathbb Q_p$? The latter is usually reserved for the $p$-adic numbers.2012-04-06

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