Timeline for Completion of a finite field extension is also finite?
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| when toggle format | what | by | license | comment | |
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| Oct 4, 2023 at 7:44 | comment | added | KConrad | Divisibility is not going to happen in general. When $K = \mathbf Q$ and $v = p$ for a prime $p$, with $w = \mathfrak p$ being a prime ideal lying over $p$ in $\mathcal O_L$, $[L_w:\mathbf Q_p] = e(\mathfrak p|p)f(\mathfrak p|p)$. Try $L = \mathbf Q(\sqrt[3]{2})$, so $[L:\mathbf Q] = 3$. Take $p = 5$. In $\mathcal O_L$, which is $\mathbf Z[\sqrt[3]{2}]$, we have $(p) = \mathfrak p\mathfrak q$ where $f(\mathfrak p|5) = 2$ and $f(\mathfrak q|p) = 1$. Thus $[L_\mathfrak p:\mathbf Q_5] = 2$ | |
| Oct 4, 2023 at 2:21 | comment | added | Yijun Yuan | Just curious, can we have a finer control on $[L_w:K_v]$? For example, can we expect that $[L_w:K_v]$ divides $[L:K]$ or something like that? | |
| Nov 8, 2015 at 18:59 | history | answered | KConrad | CC BY-SA 3.0 |