I need to show that the ideal generated by $x^2 - x + 1$ and 17 in $\mathbb{Z}[x]$ is maximal   As far as I know, it would be sufficient to show that the polynomial is irreducible mod 17, what could be a good approach to prove this?
Show that $x^2+x+4$ is irreducible over $\mathbb Z_{11}$.    The problem can be solved by putting every element from $\mathbb{Z}_{11}$ and checking if there is any root.  $0^2 + 0 + 4 = 4 \\ 1^2 + 1 + 4 = 6 \\ 2^2 + 2 + 4 = 10 \\ 3^2 + 3 + 4 = 5 \\ 4^2 + 4 + 4 = 2 \\ 5^2 + 5 + 4 = 1 \\ 6^2 + 6 + 4 = 2 \\ 7^2 + 7 + 4 = 5 \\ 8^2 + 8 + 4 = 10 \\ 9^2 + 9 + 4 = 6 \\ 10^2 + 10 + 4 = 4 $   But this process is very long one. I am seeking a short method by which I can solve such problem.   Now suppose that it is reducible, then the factors should be linear. So how can we solve the problem based on this kind of observation? Any help will be great. Thanks.
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$. I am able to successfully use wp.hooks.applyFilters() and wp.hooks.addFilter(), but I'm not able to render a React Component using wp.hooks.addAction(). However, the console.log() runs just fine within the callback for an action. This is my code:  import { createHooks } from '@wordpress/hooks';  let globalHooks = createHooks();  const Header = () =&gt; {     return (         &lt;h1&gt;Header Component&lt;/h1&gt;     ); }  const RevisedHeader = () =&gt; {     return (         &lt;h1&gt;Revised Header Component&lt;/h1&gt;     ) }  export default class App extends Component {      constructor( props ) {         super();          globalHooks.addFilter( 'replace_header', 'revisedHeaderCallback', this.revisedHeaderCallback );         globalHooks.addAction( 'after_replace_header', 'afterRevisedHeaderCallback', this.afterRevisedHeaderCallback );     };      revisedHeaderCallback( header ) {         return &lt;RevisedHeader /&gt;;     }      afterRevisedHeaderCallback() {         return &lt;RevisedHeader /&gt;;     }      render() {          return(             &lt;div className="App"&gt;                 { globalHooks.applyFilters( 'replace_header', &lt;Header /&gt; ) }                 { globalHooks.doAction( 'after_replace_header' ) }             &lt;/div&gt;         );     } } Table of contents numbering is strange Setting environment variables on OS X What is the proper way to modify environment variables like PATH in OS&nbsp;X?  I've looked on Google a little bit and found three different files to edit:   /etc/paths ~/.profile ~/.tcshrc   I don't even have some of these files, and I'm pretty sure that .tcshrc is wrong, since OS&nbsp;X uses bash now. Where are these variables, especially PATH, defined?  I'm running  (Leopard). How to sum $\sum_{x=0}^{N-1} \cos{\left( \frac{2\pi x}{N} \right)}^{2}$? I've tried solving it, but I'm not familiar with these kind of progressions and I couldn't get to a result. Also, I've found that the result is just $N/2$, but I don't get why would that be the answer, as it doesn't work for $N=1$ or $N=2$. Appreciate your help :) How to solve "Extra junk at end of file" when using apt? I'm trying to configure some proxies for a guest virtual machine and now I'm getting:  sudo apt-get update E: Syntax error /etc/apt/apt.conf.d/95proxy:4: Extra junk at end of file   The content of the file is:  cat /etc/apt/apt.conf.d/95proxy   Acquire::http::proxy  "http://10.0.0.60:3128/" Acquire::ftp::proxy "http://10.0.0.60:3128/" Acquire::https::proxy "http://10.0.0.60:3128/" Manually Installed new Python to /usr/bin, how to get Repo version back? Yes, I'm a moron, and I tried to manually install a higher point release of python 2.7.16 to /usr/bin/  (i did ./configure --prefix=/usr) on my Ubuntu 16.04 machine. It seems to work fine EXCEPT for packages that depended on python, like apt-listchanges, which fails with messages like ImportError: No module named apt_pkg  there's no make uninstall available, I've tried manually removing files, and still when I try apt --reinstall install python, Nothing gets overwritten and the python executable still gives the new 2.7.16 version...   Any idea how to get the official repository version back?? Is there a platform-agnostic and filesystem-agnostic method to obtain the full path of the directory from where a program is running using C/C++? Not to be confused with the current working directory. (Please don't suggest libraries unless they're standard ones like clib or STL.)  (If there's no platform/filesystem-agnostic method, suggestions that work in Windows and Linux for specific filesystems are welcome too.) Let $(X,d)$ be a metric space. How to prove that for any closed $A$ a function $d(x,A)$ is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof:  $$ |d(x,a) - d(y,a)| \leq d(x,y) $$ for any $a\in A$ - but we cannot just replace it by $|d(x,A) - d(y,A)|\leq d(x,y)$ since the minimum (or infimum in general) can be attained in different points $a\in A$ for $x$ and $y$, so we only have that $$ |d(x,A)-d(y,A)|\leq d(x,y)+\sup\limits_{a,b\in A}d(a,b) $$ which does not mean continuity. How to get the Input and output in a terminal to a text file? I want to create a text file which will add each command I am going to run on the terminal as a new line entry in the text file. i.e., I need to keep a record of what are the things I'm doing in the terminal. Further if I can keep a record of outputs I'm getting in the terminal, along with the inputs in the terminal, as a text file, it will be much better.   Each time I forget a command I have to search again in the Internet for the command.  Can I can make a file like above? Getting the ID of the element that fired an event Is there any way to get the ID of the element that fires an event?  I'm thinking something like:    $(document).ready(function() {   $("a").click(function() {     var test = caller.id;     alert(test.val());   }); }); &lt;script type="text/javascript" src="starterkit/jquery.js"&gt;&lt;/script&gt;  &lt;form class="item" id="aaa"&gt;   &lt;input class="title"&gt;&lt;/input&gt; &lt;/form&gt; &lt;form class="item" id="bbb"&gt;   &lt;input class="title"&gt;&lt;/input&gt; &lt;/form&gt;     Except of course that the var test should contain the id "aaa", if the event is fired from the first form, and "bbb", if the event is fired from the second form. USB WiFi dongle device that can emulate a virtual mass storage device for a home video player Is there a USB WiFi dongle device that can:   Emulate a virtual mass storage device for a home video player But in fact shows the files from a network-shared folder? Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$ I am trying to find a closed form for  $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$  It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3} \right)$$  Mathematica is unable to give a closed form for the indefinite integral.  How can we prove this result? Please help me.  EDIT  Apart from this result, the following equalities are also known to exist: $$\begin{align*} \int_0^1 \frac{\log \left( 1+x^{4+\sqrt{15}}\right)}{1+x}\mathrm dx &amp;=\frac{\pi^2}{12} \left( 2-\sqrt{15}\right)+\log \left( \frac{1+\sqrt{5}}{2}\right)\log \left(2+\sqrt{3} \right) \\ &amp;\quad +\log(2)\log\left( \sqrt{3}+\sqrt{5}\right) \\ \int_0^1 \frac{\log \left( 1+x^{6+\sqrt{35}}\right)}{1+x}\mathrm dx &amp;= \frac{\pi^2}{12} \left( 3-\sqrt{35}\right)+\log \left(\frac{1+\sqrt{5}}{2} \right)\log \left(8+3\sqrt{7} \right) \\ &amp;\quad +\log(2) \log \left( \sqrt{5}+\sqrt{7}\right) \end{align*}$$  Please take a look . Alias and functions From the bash manual     The rules concerning the definition and use of aliases are somewhat   confusing. Bash always reads at least one complete line of input   before executing any of the commands on that line. Aliases are   expanded when a command is read, not when it is executed. Therefore,   an alias definition appearing on the same line as another command does   not take effect until the next line of input is read. The commands   following the alias definition on that line are not affected by the   new alias. This behavior is also an issue when functions are executed.   Aliases are expanded when a function definition is read, not when the function is executed, because a function definition is itself a   compound command. As a consequence,   aliases defined in a function are not available until after that function is executed. To be safe, always put alias definitions on a   separate line, and do not use alias in compound commands.   The two sentences "Aliases are expanded when a function definition is read, not when the function is executed" and "aliases defined in a function are not available until after that function is executed" seem to be contrary to each other.  Can you explain what they mean respectively? How many solutions has the equation $\sin x= \frac{x}{100}$ ? How many solutions has the equation $\sin x= \frac{x}{100}$ ?   Usually when I was asked to solve this type of problem, I would solve it graphically but this one seems to be trickier. It doesn't seem wise to put $f(x)=\sin x$ and $g(x)=\frac{x}{100}$ in the same graph and then counting all the intersection points.  What would be some algebraic methods to solve this? Repeated error  I have been repeatedly encountering the following error when my laptop freezes. It may occur multiple times during the day, and should take several minutes to recover.  Source Windows  Summary Hardware error  Date ‎11/‎1/‎2018 10:11 AM  Status Not reported  Description A problem with your hardware caused Windows to stop working correctly.  Problem signature Problem Event Name: LiveKernelEvent Code:   144 Parameter 1:    3003 Parameter 2:    ffffad8c0b1a75c8 Parameter 3:    40010002 Parameter 4:    0 OS version: 10_0_17134 Service Pack:   0_0 Product:    256_1 OS Version: 10.0.17134.2.0.0.256.48 Locale ID:  1033   More details  Per my best reflection, the hardware error occurs when I am using the laptop on its docking station. Here is what I observe:   While using the laptop as normal, the "freeze" will occur and I can hardly operate the laptop: mouse cursor won't move continuously and Alt+Tab won't be able to switch across two windows. Shortly after (usually several minutes), things will go back to normal. Upon inspecting through the Performance tab in Task Manager, there had been no spike in anything, be it CPU, Memory or Disk. When I open the View reliability history tool, I will find another entry of Windows Hardware Error   Machine specification:   Thinkpad T480, i7-8650U (newly replaced module thanks to the manufacture)   With the same pair of docking stations, an older unit of T480 has had exactly the same problem.  Docking station: . Same issue has occurred on two identical docks with  slightly different monitors and accessories. OS specification: OS build 17134.376. Most recent collection of "reliability report":    Possible solutions   First, I would like to learn what exactly went wrong, hardware-wise: the manufacturer's support team did not bother too much with the precise error messages, and have ordered me to reinstall the Windows OS up to 5 times in total. [This is what they called "debugging".]   Please advise if you can decipher the recurrent "Hardware error".   Secondly, I will try to push the manufacturer to replace the docking stations, and see if the same "freezing problem' persists. Any ideas how to display a PDF file in a WPF Windows Application?     I am using the following code to run the browser but the Browser.Navigate method does not do anything!  WebBrowser browser = new WebBrowser(); browser.Navigate("http://www.google.com"); this.AddChild(browser); // this is the System.Windows.Window To run programs as sudo without password, I edited /etc/sudoers with wrong content.   Then I re-edited it with Mac's default text editor(course sudo cannot be used when /etc/sudoers is wrong).  I tried to run sudo agin, and get this error:   sudo: /etc/sudoers is owned by uid 501, should be 0 sudo: no valid sudoers sources found, quitting   What should I do? I cannot remember the root password (maybe I haven't ever set root password before). What can i put into \setmathfont{&lt;...&gt;.otf} for the unicode-math package to make it all work (with XeLaTeX or LuaLaTeX)?  There are several suggestions on the  of unicode-math, but the first option is commercial, and the second and the third do not work with the following example:  % !TEX TS-program = xelatex  \documentclass{article} \usepackage{unicode-math} \setmathfont{latinmodern-math.otf} % \setmathfont{texgyrepagella-math.otf} % \setmathfont{xits-math.otf}  \begin{document} $X\setminus Y$ \end{document}   I do not get the \setminus character in the output.  The \setminus character is also missing if i do not set a math font at all (i do not know which font is used in this case).  It works with xits-math, but i do not like it.  Is it expected by the way that the \setminus character is left blank without giving any warning? How to understand the definition of vector and tensor? Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having difficulty understanding this kind of definition. To me, it goes like this:   First we have a collection of bases or coordinate systems (do these represent reference frames?) and the transformations between them. A vector/tensor is an assignment of an array of numbers to each basis, and these arrays are related to each other by coordinate transformations.   I am wondering how coordinate systems and transformations are specified.    Do we need all possible coordinate systems and transformations to define a vector space or just a few of them? How to define the concept of basis and coordinate transformation without completing the very notion of vector?