Question is to find discriminant of polynomial $x^n-1$  I consider $f(x)=x^n-1=(x-a_1)(x-a_2)(x-a_3)\cdots(x-a_n)$  Now, $$f'(x)=[(x-a_2)(x-a_3)\cdots(x-a_n)]+\cdots+[(x-a_1)(x-a_2)\cdots(x-a_{n-1})]$$  $f'(a_1)=(a_1-a_2)(a_1-a_3)\cdots(a_1-a_n)$  $f'(a_2)=(a_2-a_1)(a_2-a_3)\cdots(a_2-a_n)$  $f'(a_3)=(a_3-a_1)(a_3-a_2)\cdots (a_3-a_n)$  and so on.. Now i need to know how many sign changes do i need to get something which looks like discriminant  I would write this in a matrix form to get some idea... $$\begin{bmatrix}12&amp;13&amp;14&amp;15&amp;\cdots&amp;1n\\21&amp;23&amp;24&amp;25&amp;\cdots&amp;2n\\31&amp;32&amp;34&amp;35&amp;\cdots&amp;3n\\\\n1&amp;n2&amp;n3&amp;n4&amp;\cdots&amp;n(n-1) \end{bmatrix}$$   See that i first row every element is in correct position i mean of the form $ij$ for $i&lt;j$  In second row only one element is odd of the form $ij$ with $i&gt;j$ but i want $i&lt;j$ in discriminant so i would change this.. So my count starts... change sign 1  In third row there are two elements which are not behaving properly... So, I should change them also.. So, Now another  two changes... On the whole $1+2$ changes...  In fourth row there would be $3$ misbehaving children so my count is $1+2+3$  In last row every body is behaving badly so i have to make $n-1$ changes in that..  On the whole i have to make $1+2+3+\cdots +n-1=\dfrac{n(n-1)}{2}$ changes..  So, $f'(a_1)f'(a_2)\cdots f'(a_n)=(-1)^{\dfrac{n(n-1)}{2}}(a_1-a_2)^2(a_1-a_3)^2\cdots (a_{n-1}-a_n)^2=(-1)^{\dfrac{n(n-1)}{2}} Disc(f)$  But then $f'(x)=nx^{n-1}$  This tells me that $f'(a_i)=n(a_i)^{n-1}$   So, $$Disc (f)=(-1)^{\dfrac{n(n-1)}{2}}  n^n(a_1a_2\cdots a_n)^{n-1}=(-1)^{\dfrac{n(n-1)}{2}}  n^n(-1)^{(n-1)^2}=(-1)^{\dfrac{n(n-1)}{2}+(n-1)^2}$$  As $(-1)^{n-1}=(-1)^{(n-1)^2}$ i would replace my $(-1)^{(n-1)^2}$ with $(-1)^{n-1}$  $$Disc (f)=(-1)^{\dfrac{n(n-1)}{2}}  n^n(a_1a_2\cdots a_n)^{n-1}=(-1)^{\dfrac{n(n-1)}{2}}  n^n(-1)^{n-1}=(-1)^{\dfrac{n(n-1)}{2}+n-1}n^n$$  i.e., $$Disc(x^n-1)=(-1)^{\dfrac{(n-1)(n+2)}{2}}n^n$$  As nobody was giving answer to my question i was trying my best and started editing this when ever i thought i find some thing and this is its final stage... This is fully solved now...  Thank you..  Any other ways of approaches would be appreciated..   Thank you :)
Let $1, \omega, \dots, \omega^{n-1}$ be the roots of the equation $z^n-1=0$, so that the roots form a regular $n$-gon in the complex plane. I would like to calculate $$ \prod_{j \ne k} (\omega^j - \omega^k)$$  where the product runs over all $j \ne k$ with $0 \le j,k &lt; n$.    My attempt so far  Noting that if $k-j = d$ then $\omega^j - \omega^k = \omega^j(1-\omega^d)$, I can re-write the product as $$ \prod_{d=1}^{\lfloor n/2 \rfloor} \omega^{n(n-1)/2}(1-\omega^d)^n$$  I thought this would be useful but it hasn't led me anywhere.  Alternatively I could exploit the symmetry $\overline{1-\omega^d} = 1-\omega^{n-d}$ somehow, so that the terms in the product are of the form $|1-\omega^d|^2$. I tried this and ended up with a product which looked like $$\prod_{j=0}^{n-1} |1 - \omega^j|^n $$  (with awkward multiplicative powers of $-1$ left out). This appears to be useful, but calculating it explicitly is proving harder than I'd have thought.  The answer I'm expecting to find is something like $n^n$.    My motivation for this comes from Galois theory. I'm trying to calculate the discriminant of the polynomial $X^n+pX+q$. I know that it must be of the form $ap^n+bq^{n-1}$ for some $a,b \in \mathbb{Z}$, and putting $p=0,q=-1$, the polynomial becomes $X^n-1$. This has roots $1, \omega, \dots, \omega^{n-1}$, so that $(-1)^{n-1}b$ is (a multiple of) the product you see above. An expression for $a$ can be found similarly by setting $p=-1,q=0$.
I can't play online. I can only play solo. I have bought Minecraft so it should work fine, but, when I try to connect to a server, it says:  Internal client error: java.net.UnknownHostException: session.minecraft.net   I have tried over 20 different servers and it says the same thing for every single one.   Note: These are not Hamachi servers. I'm attempting a direct connection. How to determine callee function name in a script To make it short, doing something like:  -bash$ function tt {  echo $0; }  -bash$ tt   $0 will return -bash, but how to get the function name called, i.e. tt in this example instead? I'm describing a situation that happened in the past. To explain it, I want to use a description that is both true now and true when the situation happened.  Specifically, I want something like:     She touched me where my neck met my collarbone.   Since my neck is still attached to my collarbone (thankfully), I'm wondering if I shouldn't use the present tense here instead:     She touched me where my neck meets my collarbone.   Which is preferable, and why? Imported CSV data is not displayed in QGIS Calculus of variations -- how does it make sense to vary the position and the velocity independently? In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat them as independent variables? What TV movie had lost space cadets who piloted yellow tripod robots, and a squid bad guy? Creating Layer in GeoServer using REST As part of a possible solution to   I want create a layer in GeoServer via REST that supplies a SQL statement along with a userid. I think i would  and publish it. I need to do this all in REST. Does anyone have any code examples? Python, Unicode, and the Windows console Prove that, for all positive real numbers $x$, $y$ and $z$, $$4(x + y + z)^3 &gt; 27(x^2y + y^2z + z^2x)$$  I've tried expressing it as a sum of squares, but haven't got anywhere.   Hints are also welcome. What is the probability that a man likes pink? In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun.  But then in physics we learn that movement is relative, and it depends on the reference point that we choose.  Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth?  Was movement considered absolute in physics back then? How to get the list of predefined colors in xcolor such that the list can be used in foreach looping? Installing an HP Laserjet 1000 I've already installed hplip, it printed a test page, but it won't print from any program. It shows the printer in the list of printers, and it's checked as the default printer. I've restarted a couple times.   When I plugged the USB cable for the printer into the computer after installing hplip, it said printer detected. Then after a couple seconds it said the printer was ready for use.   When I send something to the printer, a pop-up shows that a document was sent to the printer - but there's a triangle with an exclamation point in it next to the message. A few seconds another pop-up indicates that the printer has completed printing the job but it didn't print anything. Syntax to add multiple commands to a single command block I am looking to try and set multiple commands in a single command block, but so far everything I have tried just returns "Unknown Command".  What is the format for writing multiple commands in a single command block in Minecraft 1.8?  Can I use either of the following:  Command:(command here) Command:(command here)     /command here; /command here;   Or is there another format? Prove 24 divides $u^3-u$ for all odd natural numbers $u$ At our college, a professor told us to prove by a semi-formal demonstration (without complete induction):   For every odd natural: $24\mid(u^3-u)$   He said that that example was taken from a high school book - maybe I din't get something, but I really have no idea to prove that without using complete induction.  Any idea of smart demonstration?  Thanks for your help. (Excuse my bad English.) Debian based system, only one gui program, nothing else I would like to make a debian system where you can only open one GUI program at boot, no other graphical interface, no minimize, or any X apart from that program, is there a way to do it on debian if possible, or any other custom distro? I just want to boot open the program and allow the user to only see and use that program. Whenever I'm running or playing a game, I consistently run into an issue of setting atmosphere. I've realized the best way to do this is through more effective, engaging descriptions of settings, actions, and objects that the characters see and feel.  As a player, I feel that my actions are generally "I attack the goblin," or some variation thereof, but I'm afraid that I'll stumble over my words and further confuse the other players and I'll resort to "I attack the goblin" as to not drag the game out longer.  As a DM, my issue tends to be more with setting atmosphere with settings, locations, and objects. Effectively, the flavor texts that comes with walking into some amazing location that the characters (not necessarily the players) should be suitably impressed by. My goal here is to engage the players with some degree of spectacle, and be more inquisitive in their environment and whatever lies within it.  How can I create more descriptive, concise flavor text for the games I play in? How can I tell if an ETF is expensive? For stocks, I can easily look at the price/earnings ratio, and, according to The Intelligent Investor, stand clear of those with a ratio higher than 20.  How can I tell if the current price of an ETF is a good price to buy it? Are there such simple metrics? Why do some Jedi disappear when they die while others don't? Some Jedi disappear when they die:   In Star Wars Episode IV: A New Hope       At the end of his duel with Vader on the first Death Star, Obi-Wan Kenobi let down his guard and concentrated for a moment, disappearing entirely just as Vader's lightsaber passed through his empty robe.   In Star Wars Episode VI: Return of the Jedi      When Luke returned to Dagobah in the Return of the Jedi, Yoda also disappears when he peacefully dies.  In Star Wars Episode VIII: The Last Jedi      When Luke is exhausted after “being” on Crait he looks into the Horizon and disappears.    And others don't:   Star Wars Episode I: The Phantom Menace      When Qui-Gon Jinn's dies after his duel with Darth Maul, he doesn't  disappear and is cremated.  Star Wars Episode VI: Return of the Jedi      When Vader/Anakin Skywalker dies on the second Death Star, he does not disappear and he is cremated by Luke.      Also, they all became a "ghost" after their death. My linear voltage regulator is overheating very fast I'm using a 5 V / 2 A voltage regulator () without a heatsink. I'm testing the circuit with a microcontroller (PIC18FXXXX), a few LEDs and a 1 mA piezzo buzzer. The input voltage is aprox. 24 VDC. After running for a minute, the voltage regulator starts to overheat, meaning it burns my finger if I keep it there for more than a second. Within a few minutes it starts to smell like it's burnt. Is this a normal behavior for this regulator? What could cause it to heat that much?    Other components used in this circuit:  L1:   R2:   F1: