Let $R$ be the quadratic integer ring $\mathbb{Z}[\sqrt{-5}]$, and $I_2 = (2, 1+ \sqrt{-5})$. Show that $I_2$ is not a principal ideal in $R$.   I still find such ring "alien", hence my original attempt was to show that these 2 are co-maximal thus no single element can generate them and still remain proper ideal. I want to do this by showing G.C.D is 1.  But according to  this ring doesn't seem to be a euclidean domain, hence I can't apply euclidean algorithm here. How do I proceed from here ?  Any help or insight is deeply appreciated.
Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbb{Z}[\sqrt{-5}]$?   Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb Z[\sqrt{-5}]$. Writing $2=\alpha\beta$ in $\mathbb Z[\sqrt{-5}]$ and taking norms, $4=N(\alpha)N(\beta)$ in $\mathbb Z$, so $N(\alpha)\mid4$ in $\mathbb Z$. Similarly, since $\sqrt{-5}\in(\alpha)$ we get $N(\alpha)\mid5$, thus $N(\alpha)$ is a common divisor of $4$ and $5$, therefore $N(\alpha)=(1)$, so $\alpha$ is a unit. But that means $1\in (2,1+\sqrt{-5})=(\alpha)$, so it must be the whole ring, but it cannot to reach a contradiction, so how can I find an element, which is not in $(2, 1+\sqrt{-5})$.  Or is there a simpler method ? (Maybe not UFD would imply not PID)
Let us consider that $u\in C^2(\Omega)\cap C^0(\Omega)$ and satisfying the following equation .  $\Delta u=u^3-u , x\in\Omega$ and   $u=0 $   in  $\partial \Omega$  $\Omega \subset \mathbb R^n$ and bounded .   I need hints to find out what possible value $u$ can take ?  Thank you for ur hints in advance . How do you get \mathbb{1} to work (characteristic function of a set)? So I understand the concept of gravity, in that it's not actually a force, but more of a displacement in the spacetime grid. An object with a big enough mass will bend the spacetime, causing smaller objects to "attract" to it (not really, but good enough for what I am trying to figure out)   Ok so that makes sense, but what I am not understanding is, how come planets don't just get engulfed by the Sun? There must be something acting against the gravity for the planet to be a certain distance away from the sun. Am I not understanding this concept correctly? Simulating the Windows key on an old keyboard What is covered by CDIC savings deposit insurance? In the unlikely event that a financial institution fails in Canada, what is covered by the Canadian Deposit Insurance Corporation?  What is the limit of coverage? This is what is supposedly his claim:     My noumenal self is an uncaused cause outside of time, which therefore   is not subject to the deterministic laws of nature in accordance with   which our understanding constructs experience.   How can he say that if a noumenal self has to exist in nature and be bound to body that exists in nature? Also, if his noumenal self is free then so does 7 billion other people's, which is not the case because they all live in a societal structure where everyone has freedom that comes with a bunch conditions as to what is forbidden. I want to do:  int a = 255;  cout &lt;&lt; a;   and have it show FF in the output, how would I do this? Symbols instead of numbers as footnote markers How can I make LaTeX use symbols (*, †, ‡, and so on) instead of numbers to mark footnotes?  (The numbers are confusing because I use superscripted numbers for citations.) I'm using the alpha version of the android app on the first generation nexus 7 tablet.  The font size is a little bit small for my liking and I'm not able to pinch zoom to increase the font size.  Is this a feature that is planned in the release? How to find out why "context is incorrect"? We all know that Superman changes in the phone booth. But where does he put his regular clothes while he's on a mission? He doesn't throw it out, because he changes back. But we don't see him carrying a bag of dirty laundry while he's on a mission.  ("In his right hand, Superman has Lois Lane, and in his left hand he seems to be holding... boxers??)  And I can't imagine he leaves it in the phone booth.  So where does Superman leave his regular clothes after he changes? Efficient way to find $[\mathbb{Q}(\sqrt{5}, \sqrt{3}, \sqrt{2}): \mathbb{Q}(\sqrt{3}, \sqrt{2})]$ I want to show rigorously that this is 2. I'm sure there's a faster way than by trying to see if $\sqrt{5} = a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}$ (linear combination of the basis elements for $\mathbb{Q}(\sqrt{3}, \sqrt{2})$). I'm just not sure what. What is the difference between correlation and causation? What is the difference between correlation and causation?  Pirates and Global Temperature Example  For example, how do we know when we're dealing with correlation only and not also causation here?   What Certain Philosophers Seem to Hold Regarding This  It would seem that   Hume would saycorrelation ≡ causation. Aristotle would say that in generalcorrelation ≢ causationand thatcausation ⇔ correlationonly if we are not dealing with chance events.   ("≡" means "is identical to," and "⇔" means "implies each other.") Change hanging indent in description list I'm trying to get a description's list hanging indent to lie flush with the first line of text.  I have a description list such as the following:  \begin{description}  \item[Label the first] Lorem ipsum dolor sit amet, consectetur adipisicing                         elit, sed do eiusmod tempor incididunt ut labore et                         dolore magna aliqua.  \item[The longest label] Ut enim ad minim veniam, quis nostrud exercitation                           ullamco laboris nisi ut aliquip ex ea commodo                           consequat.  \item[Shrtlbl] Duis aute irure dolor in reprehenderit in voluptate velit esse                 cillum dolore eu fugiat nulla pariatur. \end{description}   This produces the following output:    I would like to have each paragraph indented so that the following lines are flush with the first letter of the list item's text, the way it's laid out in my TeX.  For instance, the "s" of "sectetur" should be right under the "L" of "Lorem".  I've tried looking through the documentation of , but I can't seem to find a good option.  I tried specifying [leftmargin=3cm, style=sameline], but this results in the following:    This isn't quite right: I'm trying to get each item lined up differently, so that the "L" of "Lorem" is not necessarily flush with the "U" of "Ut" or the "D" of "Duis".  Does anybody know a good way to do this? How to restrict users on changing their wallpapers? I don't want my users to change the system wallpaper. How do I the lock the wallpaper down? Update  Users are changing wallpaper by right clicking the picture and choosing set as wallpaper. How to restrict this behaviour?  Is it possible to remove Backgrounds Tab from Appearance Preferences? If yes how? Correct way to use include and includeonly when writing a large document like a thesis I am writing a large document with several chapters and I would like to work on each chapter separately. In my main.tex I include the chapters either like   \include{chap1.tex}  \include{chap2.tex}  \include{chap3.tex}   or   \input{chap1.tex}  \input{chap2.tex}  \input{chap3.tex}   I think \include is preferable if I want things to be referenced and cited correctly? but doing it the \include way latex main.tex I don't get any dvi output whereas with \input things compile fine to dvi. What's going wrong here?  I would then like to use \includeonly (or \excludeonly ) to work on individual sections of the document whilst maintaining the referencing etc.  A mwe can be handy:  \documentclass[a4paper,10pt]{article} \includeonly{chap1}  \begin{document}   \include{chap1}   \include{chap2}   \include{chap3} \end{document} Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$ Let ${\rm f}:\left[a, b\right]\to\mathbb{R}$ be twice differentiable, and suppose   $$\lim_{x\to a^{+}} \frac{{\rm f}\left(x\right) - {\rm f}\left(a\right)}{x - a} = \lim_{x\to b^{-}}\frac{{\rm f}\left(x\right) - {\rm f}\left(b\right)}{x - b} =0 $$  Show that there exists $\xi \in \left(a, b\right)$ such that $\displaystyle{% \left\vert\vphantom{\Large A}\,{\rm f}''\left(\xi\right)\right\vert \geq \frac{4\left\vert\vphantom{\Large A}% \,{\rm f}\left(a\right) - {\rm f}\left(b\right)\right\vert} {\left(b - a\right)^{2}}}$.  I don't know how to start. Any hints ?. Modern textbook on statistical field theory with an emphasis on applications to non-equilibrium phenomena? I want to be able to lock my translation/scale operations to two axes at a time, say xy, xz or yz. It currently takes two operations (one for each axis).  Example, say for a cylinder, I would like to reduce the radius. I currently have to scale on the x axis first, then on the y axis, or I have to manually input the scale values in the object properties panel.  This option is easily accessible in 3dsmax via the axis constraints command bar. Or I can click the intersecting area between the 3d manipulator of the axes in question and move in that plane. Can someone remind me of the name of the author, the name of the hero and the name of the series if any? Adventure Sci-Fi books, short by today's epic standards ( = about same length as Heinlein's shorter youth books), probably published late 50s?, early 60s? by Bantam? or Del Rey? I think at least 30, possibly more than 40 books in the series. Not comic books. The author is not a well-known writer. At least, I don't remember seeing the name on any other books, ever.   Subject: a man who can move and respond to threats extremely fast learns about a secret potion that allows a person to change bodies with someone else and thus control them. A powerful organization does not want the secret to be known, because they use the potion to control politicians on many planets. The organization's goons are people who had surgeries to remove their emotions, and function like organic computers. They can communicate over interstellar distances and follow and murder anyone who learns the secret. I remember the following detail about the potion: there are two nearly identical molecules that have to be assembled in a certain sequence. Two of the components had to be switched around and that determined who would be dominant and who would be controlled during the mind switch / body swap.   Every book finds the hero on another planet with new adventures, and usually taking another deep-sleep-passage to another world, fleeing his pursuers. The background story about the secret potion is dished out little by little in each book.  One book in the series starts like this: hero is on very hot planet, diving for valuable marine life, catches a deadly fungus disease and is rescued last moment in exchange for his catch...  One of the themes that returns often is that the hero is an excellent knife fighter.  Of course he's a womanizer. In the second (or close to 2nd) book in the series, he's involved with a woman who rules a planet.  To travel to another world, he usually takes the cheapest dead-sleep passage, and there's always a 10-25% risk you won't wake up from that.