$\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ??
$\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$
Ring of $\mathbb{Z}_2$-valued functions Let $R =\{f:\{1,2,3,4,\cdots ,10\}\to \mathbb{Z}_2 \}$ be the set of  all $\mathbb{Z}_2$ -valued functions on the set $\{1,2,3......10\}$  of the first ten positive integer. Then $R$ is a commutative ring with point-wise addition and multiplication of functions. Which of the following is true?     1) $R$ has a unique maximal ideal.      2) Every prime ideal of $R$ is also maximal.      3 ) The number of proper ideals of $R$ is $511$.      4 ) Every element of $R$ is idempotent.   R is finite commutative ring so  every prime ideal  is maximlal ideal ,so I think option  2 is true   .But for the other options   I am  helpless  . Let $G$ be a finite group and suppose there exists $f\in\text{Aut}(G)$ such that $f^2=\text{id}_G$, i.e., $f$ is its own inverse, and such that $f$ has no fixed points other than the identity $e$ of $G$, i.e., $f(x)=x\Rightarrow x=e$. Show that $G$ is necessarily abelian.   While trying to do this exercise I noticed two facts.  First, $g$ and $f(g)$ have the same order because $o(f(g))|o(g)$ and, applying $f$ again and using $f^2=\text{id}_G$, $o(f(f(g)))=o(g)|o(f(g))$ and, once the order of any element is $\geq 1$, it follows that $o(g)=o(f(g))$. Also, it's easy to see that $g$ and $f(g)$ commute.  Second, there cannot exist such an automorphism if the order of $G$ is even, because $f(e)=e$ and we can form pairs like $\{g,f(g)\}$ with $f(g)\neq g$ that are invariant under $f$, i.e., $f(\{g,f(g)\})=\{g,f(g)\}$. But once the order of $G$ is even, proceeding with the construction of the pairs, we'll end up with just one element $\neq e$, so we must have some $\gamma\in G\setminus{e}$ with $f(\gamma)=\gamma$, contradicting the hypothesis.  Let $n$ be the order of $G$ and let's fix an enumeration of the elements of $G$, say $G=\{g_1,\ldots,g_n\}$. My approach was the following. For each $\sigma\in S_n$, let $x_{\sigma}=\prod_{i=1}^ng_{\sigma(i)}$. Then we have that $f(x_{\sigma})=x_{\sigma^{\prime}}$. If it's shown that $f(x_{\sigma})=e\;\forall\;\sigma\in S_n$, then we'd have $x_{\sigma}=e\;\forall\;\sigma\in S_n$, which implies that $G$ is abelian. The problem is that $S_n$ has $n!$ elements while $G$ has $n$ elements, so there repetitions among the $x_{\sigma}$, and we cannot apply directly the reasoning of the last paragraph.  Is this the right way, or is there an easier manner to solve this? How to make applications open on the correct monitor when using multiple monitors? I'm using Windows 8.1, which has much better multiple monitor support than Windows 7 (sort of). However when I'm in a full-screen app on one monitor, and I go to the Start menu on the other and open a desktop app, it opens on the opposite monitor because of it being on that one last time the app was used.  This suspends the full-screen app and switches to the desktop view on that monitor. I then have to drag the desktop app to the other monitor which I didn't have my mouse pointer near in the first place, then switch back to the full-screen app on that monitor by going to the upper-right corner.  This is so incredibly irritating, every time I plug in my laptop to my second monitor and try to use it, this happens. Not to mention whenever I have Netflix or any kind of video in the second monitor and it just randomly stops playing when I'm trying to work on the other screen.  Most likely there is going to have to be a third-party application to install to do this, right? I would really prefer if it was free, as this was less annoying in Windows 7 when the application would only open on top of the other that was on that monitor. This is one of the maybe two annoying things I don't like about Windows 8/8.1. Everything else is fantastic, to be honest. How can I take a photo which shows the skyline and the person who is standing around six feet away from the camera with proper lighting at night?   For taking just the skyline I've used long shutter speed and it turned out great, but when including a person on it, the person becomes  blurry because it's impossible for a person to remain static for that long.   Ps.  I do have two flashes, 35mm prime lens as well as 18-300mm zoom lenses as well. The camera I use is Nikon D7200. How to get relative path from absolute path There's a part in my apps that displays the file path loaded by the user through OpenFileDialog. It's taking up too much space to display the whole path, but I don't want to display only the filename as it might be ambiguous. So I would prefer to show the file path relative to the assembly/exe directory.  For example, the assembly resides at C:\Program Files\Dummy Folder\MyProgram and the file at C:\Program Files\Dummy Folder\MyProgram\Data\datafile1.dat then I would like it to show .\Data\datafile1.dat. If the file is in C:\Program Files\Dummy Folder\datafile1.dat, then I would want ..\datafile1.dat. But if the file is at the root directory or 1 directory below root, then display the full path.   What solution would you recommend? Regex?  Basically I want to display useful file path info without taking too much screen space.  EDIT: Just to clarify a little bit more. The purpose of this solution is to help user or myself knowing which file did I loaded last and roughly from which directory was it from. I'm using a readonly textbox to display the path. Most of the time, the file path is much longer than the display space of the textbox. The path is supposed to be informative but not important enough as to take up more screen space.  Alex Brault comment was good, so is Jonathan Leffler. The Win32 function provided by DavidK only help with part of the problem, not the whole of it, but thanks anyway. As for James Newton-King solution, I'll give it a try later when I'm free. What is the difference between upstream and downstream when referring to who (or where) to go to as a developer or packager? Are there recommendations or pitfalls for playing an archer cleric? In my 4E game, a player is going to try an elven cleric with a longbow (and a mace, just in case). She chose mostly ranged powers. Are there any caveats to playing such a character, or any builds that I could recommend for her? What is the meaning of p values and t values in statistical tests? After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests.  It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results.  Many computerized tools report test results in terms of "p values" or "t values".  How would you explain the following points to college students taking their first course in statistics:   What does a "p-value" mean in relation to the hypothesis being tested?  Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value? Why does "Size on disk" vary when "Size" does not with the same set of files? Effect of linear terms on a QFT I was told when first learning QFT that linear terms in the Lagrangian are harmless and we can essentially just ignore them. However, I've recently seen in the linear sigma model, \begin{equation}  {\cal L} = \frac{1}{2} \partial _\mu \phi _i \partial ^\mu \phi _i - \frac{m ^2 }{2} \phi _i \phi _i  - \frac{ \lambda }{ 4} ( \phi _i \phi _i ) ^2  \end{equation}  with $m ^2 =-\mu^2 &gt; 0$, adding a linear term in one of the fields $\phi_N$, does change the final results as you no longer have Goldstone bosons (since the $O(N)$ symmetry is broken to begin with).   Are there any other effects of linear terms that we should keep in mind or is this the only exception of the "forget about linear terms" rule? How to start a new Linux distro? Which IP address do I use to access my server? I'm running a Samba server on my MacBook and I've also set up port forwarding to said server from my router.  My public IP is 106.51.x.x (according to google and a few other services) and my Mac's local IP is 192.168.x.x, but when I head to my router's configuration page, it says my external IP is 10.242.x.x.  This is what I don't understand, when connected to my home WiFi, using the 10.242.x.x address works just fine. But if I try to access the same over 4G, the connection times out.  As for the public address, neither WiFi nor 4G works.  Is there some setting on my Mac (I'm running MacOS Sierra) that I must alter to make it visible to the public? Single question thread with multiple questions, some duplicates I've recently run into many questions where to author asks several related questions in one post, many of them asked before.   I'm wondering what out policy is on such questions:   Should I vote-close them as being not constructive? Should I vote-close them as duplicates? Should I leave them be? Or even suggest an answer?    - Each of the questions: Waiting for an AJAX request with selenium, and headless browsing in selenium have been asked here before, but he asked both in one.  I've done my best to search for similar questions here in meta but I was unable to find any, reading appreciated. Getting nauseated from air travel, but it's not motion sickness... what to do? Connect iPhone to Mac Without Charging I am developing iOS apps and I hate, that always when I connect my iPhone to my Mac to install and test an app, the iPhone gets charged. Is there any possibility to connect my iPhone to my Mac without charging the phone?    My Environment: -  iPhone 4S (No Jailbreak) -  MacBook Air3,2 (Late 2010) Avoiding unused variables warnings when using assert() in a Release build I'm doing a masters course at a different university than I did my undergrad (both in maths). I did well in my undergrad but now I'm finding it very difficult. The shock of the change is hard to cope with.  I'm not just finding it very difficult, but I get zero feedback: we don't have any tests and I was not able to make friends so I don't know how others are finding it. I'm starting to think that I might be an idiot.   How can I build some confidence in my abilities at the same time as I rush and struggle to keep up with the massive rate of new material I need to learn? How might I get myself some reassurance that I can actually pass, or stop worrying about this so much? (I feel terribly guilty using so much money from my family to be here) It's possible that I don't really have a chance at all but knowing that would be fine. Turn off automatic preview in apple mail How to install Android OS and Android Apps on PC ? If possible , android 4.0.3 . Lazy evaluation in C++ C++ does not have native support for lazy evaluation (as Haskell does).  I'm wondering if it is possible to implement lazy evaluation in C++ in a reasonable manner. If yes, how would you do it?  EDIT: I like Konrad Rudolph's answer.  I'm wondering if it's possible to implement it in a more generic fashion, for example by using a parametrized class lazy that essentially works for T the way matrix_add works for matrix.  Any operation on T would return lazy instead. The only problem is to store the arguments and operation code inside lazy itself. Can anyone see how to improve this?