Let $\alpha=\sqrt{2+\sqrt{3}}$.  Let $L=\mathbb{Q}(\alpha)$.      Determine $Gal(L/\mathbb{Q})$   Thanks for your help.
Suppose $L=\mathbb{Q}(\sqrt{a+b\sqrt{d}})$,($d$ and $a+b\sqrt{d}$ are square free algebraic integers) when is $L/\mathbb{Q}$ a normal extension? When does $Aut(L/\mathbb{Q})=\mathbb{Z}/4\mathbb{Z}$ or $Aut(L/\mathbb{Q})=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$?
I'm working on a large c++ system that is has been in development for a few years now. As part of an effort to improve the quality of the existing code we engaged on a large long-term refactoring project.  Do you know a good tool that can help me write unit tests in C++? Maybe something similar to Junit or Nunit?  Can anyone give some good advice on the methodology of writing unit tests for modules that were written without unit testing in mind? Which is the recommended way to plot: matplotlib or pylab? How to list files that were changed in a certain range of time? How to evaluate environment variables into a string in Python? The escape velocity of earth is roughly $11 kms^{-1}$. However, what if a long ladder was built extending out of Earth's atmosphere and considerably more. Then if something was to climb up at much less than the escape velocity, what would happen when it reached the end?  And what if the object that climbed the "ladder" then fired some kind of thruster/rocket and was going fast enough so that it orbited the Earth. It would mean less energy required to get into orbit? Order of the Rubik's cube group How to remove GNOME Shell from Ubuntu 20.04 LTS to install other desktop environment from scratch? Number of non-decreasing functions? Let $A = \{1,2,3,\dots,10\}$ and $B = \{1,2,3,\dots,20\}$. Find the number of non-decreasing functions from $A$ to $B$.  What  I tried: Number of non-decreasing functions = (Total functions) - (Number of decreasing functions) Total functions are $20^{10}$. And I think there are ${20 \choose 10}$ decreasing functions. Since you choose any $10$ codomain numbers and there's just one way for them to be arranged so that the resultant is a decreasing function. However my answer doesn't match. Where am I going wrong? How can I directly compute the non-decreasing functions like without subtracting from total? ffmpeg working from command line in Terminal but not in an Automator shell script! Spacing problem at List of Figures Why must a field whose a group of units is cyclic be finite? Let $F$ be a field and $F^\times$ be its group of units. If $F^\times$ is cyclic, then show that $F$ is finite.  I'm a bit stuck. I know that I can represent $F^\times = \langle u \rangle$ for some $u \in F^\times$ and that we must have that $|F^\times| = o(u)$, where $o(u)$ denotes the order of $u$ in $F^\times$.  I tried assuming $o(u) = \infty$, but I'm not sure exactly where to go from there. I was wondering if I could get a hint. Is it: My apples and orange are/is wrong? Simple question:     My apples and orange are wrong   or     My apples and orange is wrong   I am not a native English speaker, and I am having some trouble choosing between plural are or singular is for that kind of example. OS X computer name not matching what shows on terminal My computer name in System Preferences &gt; Sharing is set to &quot;archos&quot;, but is showing as &quot;iphone&quot; on Terminal. It just started doing this after I loaded Xcode for doing iPhone development: Last login: Mon Nov  7 14:46:55 on ttys001 iphone:~ travis$   Any ideas what could be causing this? I'm going to mount my 47" LCD above my fireplace. I'm not looking for arguments against this decision. I'm looking for wiring solutions. The diagram below should help explain the situation.   _________________________________________________ |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | | p |   | m | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    #################    |     |     | |     |     |    ##             ##    |     |     | |     |     |    ##             ##    |     |     | |    P|     |    ##             ##  m |     |P C  | |_____|_____|____##_____________##____|_____|_____|    | = stud # = fireplace brick P = existing power outlet C = existing cable outlet p = new power outlet m = new media outlet   Essentially, there is an existing power outlet on either side of the fireplace and a cable outlet on the far right side of the fireplace. I'm intending to put my non-TV media equipment on the right side of the wall and the television above the fireplace. This means I need two things above the fireplace: power and HDMI. I don't want to just run an HDMI cable in the wall. I'd prefer to have some sort of conduit to simplify replacing the cable or having a second (with one of these on each end: ). I only need to drill through two studs that are 6" apart above the right side of the fireplace in order to access the vertical space where I want the other end. For the power, I need to drill through two 6"-spaced studs above the left-side of the fireplace, but then another 2 full-height studs.  There are a couple of caveats:   This is an externally facing wall, which means there's insulation. I'm assuming this will make things more difficult. I'd like to refrain from cutting more holes in the wall than is necessary for the outlets (to reduce drywall damage/repair).   I've considered a couple of options:   Cut the openings for the outlets above the fireplace, and use a flexible drill bit to drill through the studs above the fireplace. This minimizes dry-wall damage, but how difficult is this with insulation in the wall, how can I feed the conduit through those holes, and how would I drill through the full-height studs on the left for power? Cut a wide section of dry-wall (4" high?) from the desired media outlet to the right until reaching the gap above the desired lower outlet. This results in some dry-wall damage, but would allow more control for drilling holes in the studs and running conduit. This still doesn't solve the problem of getting through the full-height stud for the power. For the power, I would need to extend the cut-out two studs further.   What should I do? (Last Jedi spoilers ahead, if you haven't seen the movie, please don't read on.)  So, fridge logic moment for me:      If Vice Admiral Holdo had told Poe Dameron about the plan to just hold on long enough to make a covert run to Krait, then Poe wouldn't have felt the need to send Finn and Rose on their little side-trip, and the code-breaker wouldn't have overheard anything nor sold them out, and the First Order would've had no reason to notice the cloaked ships.   I do understand that there's a lesson in there about following orders, even if said orders come from a pretty face in a pretty dress, but that's kind of an out-of-universe reason. What was Holdo's reasoning? Surely she knew enough about Poe to realize that treating him like a mushroom1 would not result in anything good.  1 "kept in the dark and fed shit" What vintage movie had this blue floating holographic head talking to some kind of military installation? As you can see on this screenshot of 's Guake he has a startup message of ASCII art when he opens it, and that's what I'd like to do, can someone help me? Set myself as owner of /etc with chown command now getting all kinds of errors What should the order of DNS servers be for an AD Domain Controller and Why? This is a  about Active Directory DNS Settings. Related:     Assuming an environment with multiple domain controllers (assume that they all run DNS as well):  in what order should the DNS servers be listed in the network adapters for each domain controller? Should 127.0.0.1 be used as the primary DNS server for each domain controller? Does it make any difference, if so what versions are affected and how? SEDE should pick up on changes to community metadata It appears that site meta data isn't updated in SEDE. As an example, Moderators.SE was  to  in December 2014. It is still listed in SEDE as Moderators.SE, as of July 2015.  To search for Community Building data, we need to search the "Moderators" community:    It is not listed as Community Building: