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.
Cyclotomic Field Problem
2
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abstract-algebra
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3Do 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
1 Answers
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Uniqueness is easy, just use Galois correspondence (what is the Galois group of $\mathbb{Q}(\epsilon)/\mathbb{Q}$?). Now you just need to show that $\alpha$ satisfies a quadratic over $\mathbb{Q}$. Presumably you have already squared it? What do you get?
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3@Patricia Whenever you cannot see some problem to the end theoretically, try some examples. Try $p=3,5,7$, see what you find. Feel free to report back here if that doesn't help. – 2012-04-06