PS: Oops! This is somewhat duplicate. Thanks for those who made interest for this post. I will delete this post in a few hours. Please cite the following link  PS2: Now this post has a one correct answer. Should I delete this post? Let $\zeta_n \in \mathbb{C}$ be a primitive $n^{\textrm{th}}$ root of unity and call $\mathbb{Q}[\zeta_n]$ the $n^{\textrm{th}}$ cyclotomic extension of $\mathbb{Q}$.  Let $m$ and $n$ be positive integers. Describe a simple relation of $m$ and $n$ which is equivalent to $\mathbb{Q}[\zeta_n] = \mathbb{Q}[\zeta_m]$  A student solving exercises on Galois theory made the following argument. Consider $\zeta_{19}$. She could see that $-\zeta_{19}$ is one of primitive $38$th root of unity. So she conclude $\mathbb{Q}[\zeta_{19}] = \mathbb{Q}[\zeta_{38}]$. That was right. But it seems that she thought those two fields are not same. I have told her the fact. However I cannot tell a simple criterion amounts to the above question. Would you please help me?
For $n$ a positive integer, let us write $\zeta_n = e^\frac{2 \pi i}{n}$, a primitive $n$th root of unity. It is clear that, if $m$ divides $n$, then we have an inclusion of cyclotomic fields $$ \mathbb{Q}(\zeta_m) \subseteq \mathbb{Q}(\zeta_n).$$ On the other hand, these inclusions are not always strict. For example, since $\zeta_3  = \frac{1+i \sqrt{3}}{2}$ and $\zeta_6 = \frac{-1 + i \sqrt{3}}{2}$, we have $$\mathbb{Q}(\zeta_3) = \mathbb{Q}(\zeta_6) = \mathbb{Q}(i\sqrt{3}).$$ Does this sort of thing happen infinitely often, or are there just some coincidences among small numbers? If this continues, is there some way to know when a particular inclusion of cyclotomic fields is strict?
/var/log/syslog constantly logging errors, growing at 1 MB/s It seems to be the same three errors repeated:  $ tail -n15 /var/log/syslog Apr  5 23:32:01 detoo kernel: [ 6042.078785] ACPI Error: [PGRT] Namespace lookup failure, AE_NOT_FOUND (20150619/psargs-359) Apr  5 23:32:01 detoo kernel: [ 6042.078787] ACPI Error: Method parse/execution failed [\_GPE._L6F] (Node ffff8808554d42d0), AE_NOT_FOUND (20150619/psparse-536) Apr  5 23:32:01 detoo kernel: [ 6042.078788] ACPI Exception: AE_NOT_FOUND, while evaluating GPE method [_L6F] (20150619/evgpe-592)   I noticed something was wrong when I was getting low disk space warnings. I've flashed to the latest UEFI available, I've tried enabling/disabling the C states in my UEFI.  I'm running Linux Mint 17.3 Cinnamon 64-bit with kernel 4.2.0-34-generic. I have a skylake processor and z170 motherboard.  I'm quite new to linux and am at a lost as what I should do next.  EDIT: I have the same problem when booting from a USB with Ubuntu 16.04, and nearly identical error logs being constantly spammed:  Apr  6 01:01:48 ubuntu kernel: [  152.594053] ACPI Exception: AE_NOT_FOUND, while evaluating GPE method [_L6F] (20150930/evgpe-592) Apr  6 01:01:48 ubuntu kernel: [  152.594994] ACPI Error: [PGRT] Namespace lookup failure, AE_NOT_FOUND (20150930/psargs-359) Apr  6 01:01:48 ubuntu kernel: [  152.594996] ACPI Error: Method parse/execution failed [\_GPE._L6F] (Node ffff8808554e02d0), AE_NOT_FOUND (20150930/psparse-542) How to customize title page in beamer so that it looks like How can I modify title page so that it looks like it is in  video. Please seek to 16 seconds and watch title page. I do not have permission to post images. I liked this theme and I want to make a beamer theme that looks similar. I tried with beamercolorbox but I failed to position the box to the bottom.      Edit: Here is my minimum working example  \documentclass[xcolor=dvipsnames]{beamer} \defbeamertemplate*{title page}{customized}[1][] { \usebeamerfont{subtitle} \usebeamercolor[fg]{subtitle}  {\flushright \setbeamercolor{author}{bg=white,fg=Red} \usebeamerfont{author} \insertauthor\par \usebeamerfont{institute}\insertinstitute\par }  \vspace{0.2in}  \setbeamercolor{postit}{fg=white,bg=YellowOrange}  \begin{beamercolorbox}[sep=1em,wd=1.062\textwidth,ht=3cm,dp=3cm,right]{postit} \usebeamerfont{title} {\huge \inserttitle}\par \usebeamerfont{subtitle} {\insertsubtitle} \end{beamercolorbox} } \usecolortheme[named=YellowOrange]{structure} \useoutertheme{infolines} \setbeamertemplate{navigation symbols}{}  \institute{Harvard College Professor and\\ Johnstone Family Professor in Psychology, Harvard University}  \title{SAY WHAT?} \subtitle{Linguistics as a Window to\\ Understanding the Brain} \author{Steven Pinker}  \begin{document} { \setbeamertemplate{footline}{} \begin{frame} \titlepage \end{frame} } \end{document} Getting opencv error in c++ Have there been any anime with actual product placement? In Steins;Gate, Dr. Pepper is mentioned several times as the "An Intellectual Drink, For The Chosen Ones".    I've seen a lot of other shows that have stores or products with similar names to the real-life brand, so that the producers can avoid copyright issues.     Even though the inclusion of Dr.Pepper wasn't asked for by the Dr.Pepper company, after the release of Steins;Gate the sales in Japan for Dr.Pepper shot up. .   Have there been any anime where a company has paid for their product to be included in the show? Residue fields of schemes of finite type (over $\mathbb{Z}$) Suppose $X$ is a scheme of finite type over $\mathbb Z$. I want to prove that:  (1) The residue fields of closed points of $X$ are finite;  (2) For a given $q=p^n$ with $p$ prime, there is only a finite number of closed points of $X$ whose residue field is $\mathbb F_q$.  For (1), I see that one has to use some form of Nullstellensatz. First, we suppose $X=\operatorname{Spec}(A)$ affine, and $A=\mathbb{Z}[X_1,...,X_n]/I$. We need to show that if $m$ a maximal ideal of $A$, then $A/m$ is finite. Note $p=\mathbb Z\cap m$. If $p$ is maximal, we get that $A/m$ is of finite type over $\mathbb F_p$ thus a finite extension by Nullstellensatz and we are done. But $p$ could be the $(0)$ ideal in which case I neither conclude nor get a contradiction.  For (2), we can again argue locally, so we need to show that $A$ contains only a finite number of maximal ideals $m$ with $A/m$ of fixed cardinality $q=p^n$. I get an impression that to count these ideals its the same as to count the number of automorphisms of $\mathbb{F}_{q}$ which fix $\mathbb F_p$, but the corrspondence is not bijective, so we only get an upper bound which is enough. Is this idea correct? Page to Run Classes I have a urgent requirement to make a page in which list of all classes be there with the check box embedded and having button Run. If i select one or more than one classes and click on run button(Run Test) the test class associate with the particular class should run. please help me out of this requirement.  PageLayout How do I read / convert an InputStream into a String in Java? Continuity of function mapping connected set to connected set For an existing set of line features, I would like to generate a perpendicular line, of say 1km, at the midpoint of each existing line feature in ArcGIS.  I imagine that The methodology would be to:   create a tangent line at the midpoint of each existing line feature calculate the angle of the tangent line based on the angle of the tangent line, calculate the location of the endpoint of the new line create the new line with the calculated endpoint of the new line and the existing midpoint of the old line   Let's assume that we are in a UTM projection and can use coordinates to calculate distance and direction.  How would I do this with ArcGIS tools or with ArcPy? Can there be two winners at the same time in Catan? Imagine the following setup:    Player A has 6 points, including the Longest Road. Player B has 8 points with an equally long road. Player C has 9 points with a shorter road.    Now what would happen if Player C places a village, breaking Player A's road chain? At that moment Player A would have 4 points, while both Player B and Player C would have 10.   In this case, who would be declared the victor? Storage disk format to use to be utilised by all OS How can I figure out what's slowly eating my HD space? Every few days I get notices on my MacBook that it's running or run out of hard drive space. Curiously, restarting the computer will enable me to recover gigabytes of space (this past time, it was able to recover about 2.2GB). However, I can't identify anything in my personal activity that consumed that space.  It's possible that it's a rogue iTunes podcast or a huge software update that my Mac is automatically downloading - would either of these reclaim the space upon a restart?  One possibility that I can think of is that FileVault has some sort of disk leak, allocating but not freeing files. Does this make sense? Is there a tool that I can run to determine where this space is going? Assuming it is FileVault, should I try to disable it? What's the best way to turn of FileVault on a nearly full computer? How to format number with comma in apex How does the Eco-Cooler air conditioner really work? describes a device for developing countries which apparently cools the air by 5 °C without electricity. According to , it works by lowering the pressure to drag in the ambient air. Wouldn't the ambient air be at the same temperature?  Does the constriction compress the air and temporarily increase the pressure and so temperature, which then dissipates heat because it is hotter than surroundings; then after passing the constriction, the pressure returns to almost ambient but the temperature drops to below ambient?  How does the Eco-Cooler air conditioner actually work, if at all? Why bother with the dual problem when fitting SVM? How to recover a deleted file from /data partition? I have dual US and UK citizenship, but the US citizenship has only just been realized (wasn't aware I had claim to US citizenship earlier).  Unfortunately I haven't received my US passport yet. Can I still travel to the US only using my UK passport?  I understand that the US does require citizens to travel on a US passport, but would they even know that I am a dual citizen if I've only ever traveled on my UK passport? (Especially considering that my US passport has never been used before in any form - as I haven't received it).  If I was only entering the US for a small trip,  would it show on the screen that I also have US citizenship when they scan my UK passport? How to configure lightdm to allow manual logins in Ubuntu 14.04 I used to modify /etc/lightdm/lightdm.conf, adding the line greeter-show-manual-login=true, in order to allow login for hidden users and root.  Now I've just installed the new Ubuntu 14.04 and the file lightdm.conf is no longer located in /etc/lightdm. It's in /etc/init and has been totally modified.  How can I force lightdm login screen to show a field to input a login name? "Interested with the course" or "interested in the course"? Interested in/with the course   Which preposition is correct? in or with, or both. System.Exception: Too many SOQL queries: 101 I am getting a Too many SOQL queries: 101 error on a trigger when I am trying to deploy another trigger.  I am posting the trigger which is causing the error.  trigger rollupAccountShowings on Account (before insert, before update) {    if (trigger.isinsert) {       for (Account a:Trigger.new){           a.No_of_Viewings__c = 0;        }    } else {      for (account a : [select                   id,                  (select id from pb__Showings__r)                  from account                   where id in:trigger.new LIMIT 1]) {           trigger.newMap.get(a.id).No_of_Viewings__c = a.pb__Showings__r.size();      } }   Can anyone correct me where I have gone wrong and suggest me the changes.  Thanks