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I'm stuck on the proof of Theorem 4.14 in Washington - Introduction to Cyclotomic Fields.

We take a prime power $n = p^m$ and define $\pi = \zeta_n - 1$, $\zeta_n$ a primitive $n$-th root of unity. Let $\mathbb{Q}(\zeta_n)^+ = \mathbb{Q}(\zeta_n + \zeta_n^{-1})$ be the maximal real subfield. The proof claims that the $\pi$-adic valuation of $\mathbb{Q}(\zeta_n)$ only takes on even values on $\mathbb{Q}(\zeta_n)^+$. I'm not sure how to show this.

Thanks for any help.

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1 Answer 1

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The key is to notice $\mathcal{O}_{\mathbb{Q}(\zeta_n)}$ is totally ramified at $p$ and there is a unique prime of $\mathcal{O}_{\mathbb{Q}(\zeta_n)}$ which lies above $p,$ namely $\pi.$ It follows that the same holds for any intermediate extension $L$ satisfying $\mathbb{Q} \subset L \subset \mathbb{Q}(\zeta_n).$ So if $\mathfrak{p} _L$ is the unique prime of $\mathcal{O}_L$ above $p,$

$$e(\mathfrak{p}_L | p) = [L:\mathbb{Q}].$$

And thus,

$$e((\pi) | \mathfrak{p}_L) = [\mathbb{Q}(\zeta_n):L].$$

It follows

$$v_{\pi}(\mathbb{Q}(\zeta_n)^+) = v_{\pi}((\pi) \cap \mathcal{O}_{\mathbb{Q}(\zeta_n)^+})\mathbb{Z} = v_{\pi}(\mathfrak{p}_{\mathbb{Q}(\zeta_n)^+})\mathbb{Z} = e(\mathfrak{p}_{\mathbb{Q}(\zeta_n)} | \mathfrak{p}_{\mathbb{Q}(\zeta_n)^+}) \mathbb{Z} = 2\mathbb{Z}.$$

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