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I want to prove the assertion:

The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.

My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and $$ p = \epsilon^{-1} (1-\zeta_p)^{p-1} $$ where $\epsilon$ is a unit. Since $p$ is an odd prime, $(p-1)/2$ is an integer and $$ \sqrt{\epsilon p } = (1-\zeta_p)^{(p-1)/2} $$ makes sense and belongs to $\mathbb{Z}[\zeta_p]$.

How do I deal with the $\epsilon$ under the square root? I guess the condition on the congruence class of $p$ comes from that. Is this even the right way to proceed?

Uniqueness is not clear to me either. I thought about looking at the valuation $v_p$ on $\mathbb{Q}$, extending it to two possible quadratic extensions beneath $\mathbb{Q}(\zeta_p)$, then seeing how those have to extend to common valuations on $\mathbb{Q}(\zeta_p)$, but I didn't see how to make it work.

I would appreciate some help.

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    LaTeX note: `\pmod{p}` will automatically produce "$\pmod{p}$" with appropriate spacing and all.2011-03-31
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    the galois group is cyclic so there is a unique quadratic subfield2011-03-31
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    @admchrch: Related:http://math.stackexchange.com/questions/2827572016-08-20

4 Answers 4