Is the subset $\mathbb{Q}[\sqrt2]:=\{x+\sqrt{2}y\mid x,y\in\mathbb{Q}\}$ of $\mathbb{R}$ a field?
Show that the set $\mathbb{Q}[\sqrt{2}] = \{a + b \sqrt{2} \mid a, b \in \mathbb{Q}\}$ is a field with the usual multiplication and addition.
Script failing with "command not found: ^M" When I try to run the following script in zsh, via the command /bin/zsh ~/.set_color_scheme.sh I get the following error:  command not found: ^M   The script has u+x permissions and it used to work on another machine that also had zsh. Any clues why?   Note: This question is related to  (where I am trying to adapt the script for tcsh)  #!/bin/zsh # Contents of set_color_scheme.sh  export LS_COLORS=$( \ ( grep '\w' | grep -v '^#' | sed 's/#.\+//' | perl -lane 'printf "%s=%s:", shift @F, join ";", @F;' ) &lt;&lt;&lt; "  # HUMAN_FORMATTED_DATA # list one per line  # these are basic filesystem items no 00          # normal fi 00          # file di 01 34       # directory ln 00 36       # link pi 40 33       # pipe so 00 35       # bd 40 33 01 cd 40 33 01 or 01 05 37 41 mi 01 05 37 41 ex 00 91       # executable ow 01 34       # other writables   *.cmd 00 32 *.exe 00 32  # archive, compressed things etc *.gz  00 90 *.bz2 00 90 *.bz  00 90 *.tz  00 90 *.rpm 00 90 *.rar 00 90 *.zip 00 90 *.iso 00 90   *.cpio 00 31    # perl &amp; CODE *.c      33 *.h      33 *.sh     33 *.t      33 *.pm     33 *.pl     33 *.cgi    33 *.pod    33 *.PL     33 *.js     33 *.php    33 #*.xs  # strikethrough *.off 00 9 *.bak 00 9 *.old 00 9   # documents misc, html webstuff # really TEXT *.htm    94 *.html   94 *.txt    94 *.text   94 *.css    94   # MOVIE *.avi    96 *.wmv    96 *.mpeg   96 *.mpg    96 *.mov    96 *.AVI    96 *.WMV    96 *.mkv    96  # images &amp; pdf *.jpg    96 *.jpeg   96 *.png    96 *.xcf    96 *.JPG    96 *.gif    96 *.svg    96 *.eps 00 96 *.pdf 00 96 *.PDF 00 96 *.ps  00 96  *.ai  00 91 # adobe ill *.doc 00 91 # msword   # data, such as .db, .csv *.csv    95 *.dsv    95 *.db     95 *.sql    95 *.meta   95 # CONFS *.xml    95 *.yaml   95 *.yml    95 *.conf   95 # [a-z0-9]*rc ") I have this, in D1:  =sum(D3:D)   I want D1 to always be the sum of the rest of column D (D2 is a header)  I sometimes move a row, from eg 8 to 3, and that causes D1 to change to sum(d4:d).    I want D1 to stay as my sum of the column. Remote Desktop Session Black after Minimize PROBLEM: When I minimize a remote desktop session and restore it, the remote desktop screen shows up black. This only happens when connecting to a particular computer.   DETAILS: If I start clicking around in the black area, portions of the screen will start redrawing and showing up correctly. For example, if I leave a window open in the remote session and click where that window is located on the remote computer, then that window--and only that window--will redraw, and sometimes a portion of that window won't redraw (usually the toolbar).  And to clarify--the window only has to be minimized momentarily, so it doesn't seem to be a timeout issue. Clicking or typing in the remote session still causes the remote computer to respond appropriately. Disconnecting from the session and reconnecting restores the whole screen image, as does clicking all over the place in the black image (causing each section to redraw).  CONFIGURATION: This problem only happens for me when connecting to a particular computer (a W2K Server box configured to allow remote administration) and only with certain client computers. I've tried 7 different client computers with various versions of Remote Desktop (the OSes were: Win2K, Server 2003, Server 2008, Windows 7 RC, 3 XP) and two of them exhibit the problem (one is one of the XP boxes and the other is Windows 7). Those same computers can RDP to other computers without problem.  RESOLUTION ATTEMPTS: I have tried the following:   Disable the LOCAL screen saver as mentioned on  Turned off bitmap caching in the client, as mentioned on many forums. Updated to version 6.1 of the remote desktop client Using mRemote (I doubted this would work since it uses MS's code for connecting to RDP servers) Turning off all video acceleration.   QUESTION: Any ideas on what is causing this? Is there a quicker way to turn on WiFi Hotspot? is there a way to automate (or at least reduce the number of steps) to turn on or turn off the WiFi hotspot on my Galaxy 3.  I have to    unlock  press the menu button  press settings press More Settings Tethering and portable hotspot Slide portable hotspot on. What is the fastest algorithm for finding the natural logarithm of a big number? Make a div fill the height of the remaining screen space I was trying to create an AnyDice function to model the Great Weapon Fighting fighting style (which lets you reroll 1s and 2s), but I couldn't get it to work on any arbitrary dice.  I've found this one:  function: reroll R:n under N:n {    if R &lt; N { result: d12 } else {result: R} } output [reroll 1d12 under 3] named "greataxe weapon fighting"   And it works fine. But I don't know how to make the function generic so i don't need to change the d12 every time i want a different dice to reroll.  I've tried  function: reroll R:n under N:n {    if R &lt; N { result: d{1..R} } else {result: R} } output [reroll 1d12 under 3] named "greataxe weapon fighting"   but it is not giving the right probabilities. Maybe if I could fetch the die size inside the function... allow you to create code which defines the methods of classes that implement it. You cannot however add any code to those methods.    allow you to do the same thing, along with adding code to the method.   Now if you can achieve the same goal with abstract classes, why do we even need the concept of interfaces?   I've been told that it has to do with OO theory from C++ to Java, which is what PHP's OO stuff is based on. Is the concept useful in Java but not in PHP? Is it just a way to keep from having placeholders littered in the abstract class? Am I missing something? Using Unicode Zero Width Spaces allows all space comments How many "super imaginary" numbers are there? How many "super imaginary" numbers are there? Numbers like $i$? I always wanted to come up with a number like $i$ but it seemed like it was impossible, until I thought about the relation of $i$ and rotation, but what about hyperbolic rotation? Like we have a complex number $$ z = a + bi $$ can describe a matrix $$ \begin{bmatrix} a &amp; -b \\ b &amp; a\end{bmatrix} $$ You can "discover" $i$ by doing (which is used for another discovery) $$ \begin{pmatrix} a \\ b \end{pmatrix} \cdot \begin{bmatrix} c &amp; -d \\ d &amp; c\end{bmatrix} = \begin{pmatrix} ac - bd \\ ad + bc \end{pmatrix} $$ $$ (a + bi) \cdot (c + di) = ac + adi + bci + bdi^2 $$ From here on you can infer that $ i^2 = -1 $.  So what if we do the same thing, but a different matrix? $$ z = a + bh$$ can describe a matrix $$ \begin{bmatrix} a &amp; b \\ b &amp; a\end{bmatrix} $$ and we can discover it the same way $$ \begin{pmatrix} a \\ b \end{pmatrix} \cdot \begin{bmatrix} c &amp; d \\ d &amp; c\end{bmatrix} = \begin{pmatrix} ac + bd \\ ad + bc \end{pmatrix} $$ $$ (a + bh) \cdot (c + dh) = ac + adh + bch + bdh^2 $$ From here we infer that $ h^2 = 1 $.  Also $$ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots  $$ $$ \begin{align} e^{xh} &amp; = 1 + \frac{xh}{1!} + \frac{(xh)^2}{2!} + \frac{(xh)^3}{3!} + \frac{(xh)^4}{4!} + \frac{(xh)^5}{5!} + \cdots \\ &amp; = 1 + \frac{xh}{1!} + \frac{x^2}{2!} + \frac{x^3h}{3!} + \frac{x^4}{4!} + \frac{x^5h}{5!} + \cdots \\ &amp; = \cosh{x} + h \cdot \sinh{x} \end{align} $$  How many more numbers like this are there? And does that mean that for each set of trigonometric functions there exists a number which can turn multiplication into a rotation using those trigonometric functions?  (Sorry if I got some things wrong) Is there a word to describe the virtue of "to do something and ask for nothing in return"? I am wondering if there is a positive, formal counterpart to the slang "No strings attached" to describe, for example, a parent's unquestioned devotion to their children, or the virtue of someone's "a friend in need is a friend in deed", help first, question later mentality. No matter where I look, I just can't find gnome number [insert number here]. Where can I find every gnome in the game? Why Doesn't the Adamantium Bone Covering Kill Wolverine? I've seen the  here before, as well as on many sites on the web.  I've also seen different info on what was covered.  I'm not a huge X-Men fan, but I do enjoy the characters and stories, so I know I'm probably missing out on a lot, and may have missed the obvious along the way because I didn't read a particular series or something like that.  I had a relative die from leukemia a few years ago and that reminded me of basic junior high science: that red and white blood cells are produced in the bone marrow and must be able to reach from the marrow into the blood vessels.  The leukemia my relative had was the kind where the "bad" cells would surround the marrow and prevent the new "good" blood cells from getting out to the rest of the body.  So just how much of Wolverine's skeleton is covered with Adamantium?  If it's the whole thing, there'd have to be openings for new red blood cells to get out if he is to remain alive.  I've never seen a reference to this issue, is it ever addressed?  That also raises another issue along the same line: Are hard to reach areas, such as ball and socket joints covered?  If not, then stress on other covered areas could be transferred to the weaker uncovered areas, which would lead to breakage there.  And if joints like that are covered, is there anything to say how it was done?  Are there openings or gaps in the Adamantium covering on his body?  Are there any references to any of these issues that deal with them? Make apt-get (or aptitude) run with -y but not prompt for replacement of configuration files? When running apt-get -y install &lt;packages ...&gt; on Ubuntu 10.04 I would like apt-get (or aptitude if that makes it easier) to not prompt me when installing additional dependencies (behavior of -y as I understand it) and but not prompt me about overwriting configuration files, instead assume to keep the existing ones always (which is usually the default). Unfortunately --trivial-only seems to be the inverse of -y and not affect the prompt that is shown, according to the man page.  In particular packages auch as samba, nullmailer, localepurge and lighttpd have forced me to interact with the terminal, even though the whole procedure was scripted and meant to be non-interactive. How to perform a full system dump/backup with a broken screen? So, yesterday my phone took a serious tumble.  It is now essentially useless, as the screen either fails to turn on, or when it does, there is nothing useful to see (just lines of primary colors, mostly).  I believe (from the fact that it still chirps at me when emails come in, etc), that the phone is still functional, otherwise.  I would like to perform a full system dump, so that I can get off any files from the phone (a TMobile SGS2 (hercules) running slimrom 4.3 build 1).  The sdcard probably isn't important, as I can just pull it, but the phone's own internal storage I should like to get a copy of.  When I plug the phone into a usb port, it 'mounts' the usb via MTP, and I can see all the files.  Unfortunately, when I do a select all | copy | paste operation, the paste op dies before transferring all files.  I've tried a couple other ADB shell operations, in hope of flipping regular USB mounting on, but nothing has worked.  Any ideas how I can get a full dump of my poor phone's contents?  Edit  Currently running a adb pull /sdcard/ ... Its copying files presently, but its too early to tell if its going to get evertying I'm expecting.  Edit #2  looks like its going to work...its taking a rather long time, but it appears to be grabbing everything I was expecting it to. I'm trying to install Windows 7 with a 4GB flash drive, but my error that comes up is that my hard drive I'm trying to install on is in ext4. I need to format it to read NTFS. I can't seem to find any topics on how to format an active hard drive. I found a topic that explains how to move Ubuntu to a new drive, but it's a bit confusing to me. Google Cloud frequently doesn't have enough resources How do I check if a list is empty? For example, if passed the following:  a = []   How do I check to see if a is empty? biblatex, biber, is it possible to \printbibliography between two years? With biblatex and Biber, is it possible to print all the bibliography (\printbibliography) from (let's say) 2003 to 2008 in one section? And then in another section print all the bibliography from 2009 to 2016? If the order of all nontrivial elements in a group is 2, then the group is Abelian. I know of a proof that is just from calculations (see below). I'm wondering if there is any theory or motivation behind this fact. Perhaps to do with commutators?    Proof: $a \cdot b = (b \cdot b)  \cdot (a \cdot b) \cdot (a \cdot a) = b \cdot (b \cdot a) \cdot (b\cdot a) \cdot a = b \cdot a$.