Infinite perfect field of characteristic p
Can we have a perfect field of characteristic $p$ with a non-archimedean valuation?
I need to find all group homomorphisms of $S_n\to\mathbb{C}^*$ (for $n\ge2$), and also all group homomorphisms of $A_n\to\mathbb{C}^*$ (for $n\ge2$).  Someone on this site, two years ago, seems to have already asked the same question (). I understand what the questioner says there, but I don't understand it when he says:     Therefore there is exactly one $g:\mathbb{Z}/2\mathbb{Z}\to\mathbb{C}^*$, so all elements of $a\in\mathbb{C}^*$ give us $\operatorname{ord}(a)\mid g[\mathbb{Z}/2\mathbb{Z}]$, so $a$ can only be $-1$ or $1$. Now we know that $f:S_n\to\{-1,1\}$ &hellip;   What is he saying there? And how to go on from there? (I do not understand the answer given either &hellip;)  I hope that someone here can help me further. (And if I'm not allowed to ask this question because is has already been asked two years ago, please tell me what to do instead).  Thank you in advance! Numbers to the Power of Zero Can I play against or with Xbox 360 players through my Xbox One? 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Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$ How to set up Dolphin as default file manager? Make a momentary switch control a toggle What are the simplest, cheapest, smallest ways to make a momentary switch produce a 2-state toggling output (latching momentary switch)?  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Clarification following close votes from people who think this question is primarily opinion-based: By "OK" I mean the sense of "legally OK," "officially accepted," "instantiated as policy," "sanctioned by authorities," or "OK according to the regulations and policies and laws and whatever other formal rules govern forced removal from flights due to overbooking." Whether or not you personally consider it socially acceptable does not matter as an answer, unless you have formalized that into a citeable adopted form that governs what happens and want to explain the reasoning behind that rule. Meaning of convolution? I'm downloading GTA V. After 20gb / 100gb I paused my download for restarting my pc. Then download restarts from 0%.  Why this could happen? Can I restart my PC during downloads without losing progress? How can I change the page URL without refreshing the page? How to disable Alt + ` (backtick/grave) Where to ask "subjective and argumentative" questions on programming? How to resolve "iTunes could not back up the iPhone because the backup was corrupt"? When I try to sync my iPhone with my computer, I get the following message:       iTunes could not back up the iPhone "iPhone Name"   because the backup was corrupt or not compatible   with the iPhone.      Delete the backup for this iPhone, then try again.   Yet, here is no option to delete the backup. Where are backups stored / how can I delete the backup?  This is iTunes version 10.5.3 running under  Mac OS X 10.6.8.