My understanding is that for $\mathbb{Q}(3^{1/3},3^{1/4})$ to be a Galois extension of $\mathbb{Q}$ there must exist a polynomial over $\mathbb{Q}$ with roots $3^{1/3},3^{1/4}$ that splits in $\mathbb{Q}(3^{1/3},3^{1/4})$. Is it true then that this field is not a Galois extension in this case as the field does not contain $i$ (since some roots of $x^3-3$ and $x^4-3$ are complex) or am I missing the point?   Thanks.
From a previous question, we have that $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[4]{3})$, and I am assuming that we would need to use this to justify the answer to the question.   Would it be right to use some idea of a splitting field here? Or is it something to do with $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ not containing a primitive 3rd root of unity?   This is a past exam paper question worth 8 marks, so it doesn't seem like it's a few lined answer, but I am a little stuck on where to go.
.NET - What's the best way to implement a "catch all exceptions handler" I'm wondering what the best way is to have a "if all else fails catch it".  I mean, you're handling as much exceptions as possible in your application, but still there are bound to be bugs, so I need to have something that catches all unhandled exceptions so I can collect information and store them in a database or submit them to a web service.  Does the AppDomain.CurrentDomain.UnhandledException event capture everything? Even if the application is multithreaded?  Side note: Windows Vista exposes native API functions that allow any application to recover itself after a crash... can't think of the name now... but I'd rather not use it, as many of our users are still using Windows XP. How to expand the ext4 primary partition size in a VMware Player virtual disk? I have a VMware Player virtual machine setup, running Ubuntu 10.04 as the a guest OS.  The VM was created on a 20GB virtual drive, with a single partition for working space. Finally I got to the point where I need to expand the size of the virtual drive. Resizing the drive itself, adding some 10GB, was easy via the VMware Player menus. I now need to resize the main partition itself (I prefer this over creating a separate partition and mounting it).  According to GParted, I now have the following partitions:    My question is, how can I increase the size of the Filesystem partition to include the free 10GB?  From information I found online, it looks like one limitation is the fact that he swap partition is immediately adjacent to the filesystem, so my understanding is that first thing would be to move that partition to the end of the volume. How can this be done?  Also, some sources suggest that the system should be booted from a LiveCD, so the filesystem is not the boot partition, and thus make it easier to resize. As mentioned above, I run a VM, so I am not sure how to boot it from a LiveCD. Any ideas? Can I print a custom paper on standsard size in word? In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational,  and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but I am curious as to whether anyone has a method for proving this without making use of the Taylor Series Expansion. I feel as though doing so must be possible using some number theory, but am low on ideas as to an alternative approach to the question.  Note:  If anyone is interested in my solution using Taylor Series Expansion (although it is not the focus of my question), here it is :      From    $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} -  \dots$$   We see that    $$ \alpha = \sin 1 = 1 - \frac{1^3}{3!} + \frac{1^5}{5!} -  \dots $$   Given integers $a$ and $b$, if $\alpha = \frac{a}{b}$ then it follows that $b!\alpha \in \mathbb{Z}$, and $b!\alpha = C + D$ where $C \in \mathbb{Z}$ and we have :   $$ D = \begin{cases} \pm(\frac{1}{b+1} - \frac{1}{(b+1)(b+2)(b+3)} + \dots) \text{    if $b$ is even}\\ \pm(\frac{1}{(b+1)(b+2)} - \frac{1}{(b+1)\dots(b+4)} + \dots ) \text{ if $b$ is odd}\\ \end{cases} $$   In each case we can see that $0 &lt; D &lt; 1$, giving us a contradiction.    Thus, we have that $\sin 1 $ is irrational.    $$ \blacksquare $$ How does a pharmacy discount card work? What happens if my shield breaks? What are the alternatives to the FHS? I'm a long time Linux user for over 15 years but one thing I hate with a passion is the mandated directory structure. I don't like that /usr/bin is the dumping ground for binaries or libs in /usr/lib, /usr/lib32, /usr/libx32, /lib, /lib32 etc... Random stuff in /usr/share etc. It's dumb and confusing. But some like it and tastes differ.  I want a directory structure where each package is isolated. Imagine instead if the media player dragon had it's own structure:  /software/dragon /software/dragon/bin/x86/dragon /software/dragon/doc/README /software/dragon/doc/copyright /software/dragon/lib/x86/libdragon.so   Or:  /software/zlib/include/zlib.h /software/zlib/lib/1.2.8/x86/libz.so /software/zlib/lib/1.2.8/x64/libz.so /software/zlib/doc/examples/... /software/zlib/man/...   You get the point. What are my options? Is there any Linux distro that uses something like my scheme? Can some distro be modified to work like I want it (Gentoo??) or do I need LFS? Is there any prior art in this area? Like publications on if the scheme is feasible or unfeasible?  Not looking for OS X. :) But OS X-inspired is totally ok.  Edit: I have no idea how PATH, LD_LIBRARY_PATH and other environment variables that depend on a small set of paths should work out. I'm thinking that if I have the KDE editor  installed in /software/kate/bin/x86/bin/kate then I'm ok with having to type the full path to the binary to start it. How it should work for dynamic libraries and dlopen calls, I don't know but it can't be an unsolvable engineering problem. How do I troubleshoot high 'svchost.exe' usage in Windows 7? I'm having a problem with Windows 7 64-bit. I thought it was slow and all, but then I saw that the CPU usage was always around 80% and started searching for a solution.  There are two svchost.exe's consuming around 30% each and in the resources monitor there's a system interrupts consuming 45% all the time. I tried closing the applications, but it makes no difference.  I tried some other things that I've found on Google, like disable system updates, but it didn't work.  I don't know if it will help but here's my specifications:  Core 2 Duo 4400 ATI Radeon 4850 4 GB DDR2 RAM     EDIT  I ran the suggested program and got this information; did I get it right?      EDIT  As you asked here it is, did I get it right now? The other TCP/IP there's nothing.    EDIT  I ran msconfig and took the services that one of the svchost.exe processes was using out of the startup and now my CPU is around 50%, but I still would like to improve it further. I can't lose that much CPU power just because of Windows...    EDIT  Yeah, there's nothing I can do here. I am going to reinstall Windows&nbsp;XP soon, it's really weird... Before upgrading to macOS Sierra, I really liked the Character Accent Menu by doing a long press on a key (for example, the character e) and being able to add an accent on it.   However, this functionality seems to not work anymore, and so far, I can't find out why?   Also, before, doing a long press on the delete button, would repeatly delete characters. Now, it is doing only one character deletion disregarding how long I'm pressing the key. So if I want to erase 10 characters, I need to press the delete button 10 times...  Also, I am not able to log off or shut down. When doing it, all applications are closed, status bar disappears but nothing happens. The dock remains here. I have no way to shut down or log out without doing a long press on the power button.  Is someone facing the same issue?  EDIT : Logging with a new user account or guest account fix all the problems. But I still would like to know what, in my user account, became corrupted during the upgrade. Allow diamond moderators to reverse question migrations? It has happened on occasion, due to the high amount of 10K moderators on SO that questions are migrated to the wrong site, and needs to be reversed. Although currently it is possible to re-open the question on the original site and delete the one on the destination site, it still leave the migrated too banner behind.  It would be useful if diamond moderators could reverse a migration, removing it from the destination site completely and re-activating it on the original site and at the same time removing the banners.  Previously it has been discussed educating the users around when to move questions, however this doesn't stop the problem.  As per Shog9  I have to admit I do like Shog9's suggestion regarding the post being closed as not programming related rather then re-opened, however, reasons for closing it on other sites should also be kept in mind, as this also applies to stuff being sent back forth between the sites. Why is the product of all units of a finite field equal to $-1$? Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by $a$. Then the product in question can be written as  $$ a_1\cdots a_{q-1}=1\cdot a\cdot a^2\cdots a^{q-2}=a^{(q-2)(q-1)/2}. $$ However, I'm having trouble jumping from that product to $-1$. What is the key observation to make here? Thanks! Batch Requests/sec reported by DMV millions of times larger than Activity Monitor I have verified through SQL Server Profiler that my request per-second are around 30 request/s as is corroborated by SSMS Activity Monitor but sys.dm_os_performance_counters is reporting hundreds of millions/s.  Any idea what might be causing this gross discrepancy?  Query:  SELECT     RTrim(LTrim(object_name)) as object_name,     RTrim(LTrim(counter_name)) as counter_name,     cntr_value FROM     sys.dm_os_performance_counters  WHERE      instance_name IN ('', '_Total')     and counter_name IN (         N'Batch Requests/sec'         , N'SQL Compilations/sec'         , N'SQL Re-Compilations/sec'         , N'Transactions/sec')   Results:  object_name counter_name    cntr_value SQLServer:Databases Transactions/sec    191721399 SQLServer:SQL Statistics    Batch Requests/sec  242955426 SQLServer:SQL Statistics    SQL Compilations/sec    42048371 SQLServer:SQL Statistics    SQL Re-Compilations/sec 1200947 I’m a Filipino citizen. My flight goes from Vancouver to Manila via Tokyo Haneda. I will arrive in Japan at 6pm then depart at 9am the next day. I'm not planning to go out of the airport during my layover, so I don't see a need to have a transit visa. However, I called the Consulate-General of Japan here in Canada, and they asked me to get a transit visa.  Then I called the Ministry of Foreign Affairs in Japan and asked them the same question.. at first, they told me I don’t need a visa as long as I stay inside the airport, but when I told them that the Consulate-General of Japan in Canada asked me to get a visa, they also asked me to get one.  I'm so confused. I would like to know your knowledge or experience regarding this. Having trouble adding objects to NSMutableArray in Objective C Why does a passport have a place of birth field on it? Almost all passports I have seen have a place of birth field on them. For instance, on the US Passport,    Is there a specific reason to this? How to tell Google that I have changed my website URLs? Is there any way to import a .json model into Blender? If I run:  sudo apt-get --yes install postfix    or  sudo bash -c 'yes | apt-get --yes install postfix'   an interactive prompt appears to configure postfix. I need to automate the installation of postfix (I can modify the configuration files after the install).  Is there some magic that lets me install postfix (and other package) without human intervention? What did Ken Thompson mean when he said, "I'd spell creat with an 'e'."? What is Ken referring to? Is there a "creat" command? Bottom of tool shelf grayed out I have lost functionality in the tool shelf. When I click on a selector or something in the bottom part of the tool self that you can hide and unhide, nothing happens. I can see the options but when I click it doesn't work. See screenshot where it says "Add Cylinder".