Given is the polynomial $\varphi(X)=X^4+1$.  Now there are two tasks:  (1) Show, that $ \varphi(X)$ is reducible in $\mathbb F_p [X]$, where $p$ is prime number with  $p \equiv1$ (mod 4).  (2) Show, that $ \varphi(X)$ is reducible in $\mathbb F_p [X]$, where $p$ is prime number with  $p \equiv3$ (mod 4).  My ideas until now:  (1) I try to find $a,b \in \mathbb F_p [X]$, such that $X^4+1 = (X^2+a)(X^2+b)$. Not to use the middleterm $+cX$ was a hint. Now $(X^2+a)(X^2+b)=X^4+aX^2+bX^2+ab=X^4+(a+b)X^2+ab$. This should equal $X^4+1$, so I know, that $a+b \equiv 0$ mod p and $ab \equiv 1$ mod p. This means, that $a$ and $b$ must be units in $\mathbb F_p$, more exactly, $a$ is the multiplicative inverse element of $b$ in $\mathbb F_p$. $a+b \equiv 0$ reasons, that $a$ is the additive inverse to $b$.  But know I don't know, how I can go on with conclusion. I hope, somebody can help me.
How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$?   For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not solutions of $x^4+1=0$ then if it is reducible the factors are of degree $2$. This gives that $x^4+1=(x^2+ax+b)(x^2+cx+d)$ and solving this system of equations $\bmod 3$  gives that $x^4+1=(x^2+x+2) (x^2+2x+2) \pmod 3$. But is there a simpler method to factor $x^4+1$ modulo a prime $p$?
.NET String.Format() to add commas in thousands place for a number Maximum total distance between points on a sphere What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . Behaviour of printf when printing a %d without supplying variable name I've just encountered a weird problem, I'm trying to printf an integer variable, but I forgot to specify the variable name, i.e.  printf("%d");   instead of  printf("%d", integerName);   Surprisingly the program compiles, there is output and it is not random. In fact, it happens to be the very integer I wanted to print in the first place, which happens to be m-1.  The errorneous printf statement will consistently output m-1 for as long as the program keeps running... In other words, it's behaving exactly as if the statement reads  printf("%d", m-1);   Anybody knows the reason behind this behaviour? I'm using g++ without any command line options.  #include &lt;iostream&gt; #define maxN 100 #define ON 1 #define OFF 0  using namespace std;  void clearArray(int* array, int n); int fillArray(int* array, int m, int n);  int main() {     int n = -1, i, m;     int array[maxN];     int found;      scanf("%d", &amp;n);      while(n!=0)     {         found=0;         m = 1;         while(found!=1)         {             if(m != 2 &amp;&amp; m != 3 &amp;&amp; m != 4 &amp;&amp; m != 6 &amp;&amp; m != 12)             {                 clearArray(array, n);                 if(fillArray(array, m, n) == 0)                 {                     found = 1;                 }             }             m++;         }          printf("%d\n");          scanf("%d", &amp;n);     }      return 0; }  void clearArray(int* array, int n) {     for(int i = 1; i &lt;= n; i++)         array[i] = ON; }  int fillArray(int* array, int m, int n) {     int i = 1, j, offCounter = 0, incrementCounter;      while(offCounter != n)     {         if(*(array+i)==ON)          {             *(array+i) = OFF;             offCounter++;                }         else          {             j = 0;             while((*array+i+j)==OFF)             {                 j++;             }             *(array+i+j) = OFF;             offCounter++;                    }         if(*(array+13) == OFF &amp;&amp; offCounter != n) return 1;         if(offCounter ==n) break;          incrementCounter = 0;                while(incrementCounter != m)         {             i++;             if(i &gt; n) i = 1;             if(*(array+i) == ON) incrementCounter++;          }            }      return 0; } Sci-Fi Movie/Show Where a Soul is Captured in a Glass Vessel SQL Coalesce in WHERE clause I'm trying to implement optional parameters in a stored procedure that I have but I am running into a problem. Here's a simplified query to illustrate the issue:  SET ANSI_NULLS OFF  DECLARE @MiddleName VARCHAR(20); SET @MiddleName = NULL;  SELECT * FROM [Customer] WHERE [LastName] = 'Torres' AND [MiddleName] = COALESCE(@MiddleName, [MiddleName])   When I run this query I need to get one row back because one Torres has NULL in the [MiddleName] column. But the query returns zero rows. Using IFNULL() produces the same result. From researching COALESCE, I was under the impression that NULL would be returned if all expressions are NULL. As I am not a SQL expert I assume that I am missing something, but what is it.....  Thanks in advance for any help. My Android device screen got broken and I can't see what I'm doing. That's the only wrong thing with my device, the rest is apparently working OK. Is there any way to send my device screen to my computer? I don't want to work blindly anymore! Generating a unique machine id Or, an introductory guide to robust filename handling and other string passing in shell scripts.  I wrote a shell script which works well most of the time. But it chokes on some inputs (e.g. on some file names).  I encountered a problem such as the following:   I have a file name containing a space hello world, and it was treated as two separate files hello and world. I have an input line with two consecutive spaces and they shrank to one in the input. Leading and trailing whitespace disappears from input lines. Sometimes, when the input contains one of the characters \[*?, they are replaced by some text which is is actually the name of files. There is an apostrophe ' (or a double quote ") in the input and things got weird after that point. There is a backslash in the input (or: I am using Cygwin and some of my file names have Windows-style \ separators).   What is going on and how do I fix it? I need to check CPU and memory usage for the server in java, anyone know how it could be done? Harry was never a Horcrux of Voldemort     I have heard this line on SE a lot &amp; also vaguely read about that discussion.     My question is if Harry was not Voldemort's Horcrux &amp; if all the other 6 (5 inanimate objects &amp; Nagini) were destroyed beyond repair &amp; Harry tries to Avada Kedavra Voldemort, would he die? What would happen to Voldemort's soul inside Harry.  I've asked this question keeping in mind Harry isn't a technical Horcrux but only a part of Voldemort's soul is in him. So can Voldemort resurrect from this soul?    If its a duplicate, please mention the relevant discussion link in the comments Join lines from two files Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties. Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties:  (1) $f$ has derivative of all orders, and  (2) for all $x,y \in \mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.  Which of the following sentences is true?  (a) Any $f\in A$ is a polynomial of degree less than or equal to 1  (b) Any $f \in A$ is a polynomial of degree less than or equal to 2  (c) $\exists f \in A$ which is not polynomial  (d) $\exists f \in A$ which is a polynomial of degree 4  It is a problem from TIFR GS-2018. I have found polynomials upto degree 2 hold this properties but have to exclude the possibility of (c) and (d). I also found out if $f'(x)$ has at most one real root and clearly graph of $f$ is symetric about that root. How can I proceed further? I'm moving all files to a new computer, via an external hard drive. I am certain that broken links will be a concern. What can I do to fix this? Is there a 'Best Practices' kind of way to prevent this problem beforehand? I have many hundreds of .blend files, and of course, I fear losing them. Best way to remove an event handler in jQuery? I have an input type="image". This acts like the cell notes in Microsoft Excel. If someone enters a number into the text box that this input-image is paired with, I setup an event handler for the input-image. Then when the user clicks the image, they get a little popup to add some notes to the data.  My problem is that when a user enters a zero into the text box, I need to disable the input-image's event handler. I have tried the following, but to no avail.  $('#myimage').click(function { return false; }); So, the question pretty much says it all. I'm on Snow Leopard, and I do a lot of web development, particularly in Rails 3 which makes heavy use of the console.  I've seen some notable bloggers etc. mention Zsh as their preference over Bash, but I don't know what difference it would make.  Could anyone give me a good comparison of what difference there is and what might make one prefer one option or the other? Taylor's Remainder $x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}<\log(1+x) <x-\frac{x^2}{2}+\frac{x^3}{3}$ Prove that   $\displaystyle x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}&lt;\log(1+x) &lt;x-\frac{x^2}{2}+\frac{x^3}{3}$  My attempt: I can prove this by taking one side at a time and assuming $\displaystyle f(x) = \log (1+x)-x+\frac{x^2}{2}-\frac{x^3}{3}$   and then proving that it is a decreasing function and then the same for the other side.  But I am looking for a better solution using Lagrange's Mean Value Theorem or using Taylor's remainder. Variable wavelength IR Diode? I was thinking of building a DIY IR Spectroscope but for this, I need to find an IR diode that can do variable wavelengths. So far I have looked at Farnell and Mouser but all the diodes I have looked at seem to be fixed on a wavelength.   Is this possible? or would I need some sort of filters and optical equipment, if so which parts would I need?  Thanks. Multicollinearity in Decision Tree Can anybody please explain the affect of multicollinearity on Decision Tree algorithms (Classification and regression). I have done some searching but was not able to find the right answer as some say it affects it and others say it doesn't. I'm looking for some sample code that will sort the list items in an HTML list by alphabetical order.  Can anyone help?  Here is a sample list for people to work with:  &lt;ul class="alphaList"&gt;     &lt;li&gt;apples&lt;/li&gt;     &lt;li&gt;cats&lt;/li&gt;     &lt;li&gt;bears&lt;/li&gt; &lt;/ul&gt; Tools to get a pictorial function call graph of code I have a large work space which has many source files of C code. Although I can see the functions called from a function in MS VS2005 using the Object browser, and in MSVC 6.0 also, this only shows functions called from a particular function in a non-graphical kind of display. Additionally, it does not show the function called starting from say main(), and then the functions called from it, and so on, deeper inside to the leaf level function.  I need a tool which will give me a function call graph pictorially with functions callee and caller connected by arrows or something like that, starting from main() to the last level of function, or at least showing a call graph of all functions in one C source file pictorially. It would be great if I could print this graph.  Any good tools to do that (need not be free tools)?