Why is $ℤ[X]/(X^2-1)$ not isomorphic with $ℤ ×ℤ$ ?   I understand why this is true in the case of $\mathbb{Q}$. But $(X-1)+(X+1) ≠ℤ[X]$, so therefore I can't use the same reasoning. I don't see how I can proof that there can't be an isomorphism.
I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$.    I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be looking for elements which have inverses in $\mathbb{Z}[X]/(X^2-1)$ and hopefully find more or less than 4. But I didnt succeed, so I need hints. Thanks.
"State-of-the-art" and "technology" How to close entire application when application has more than 2 forms without showing messagebox more than once Update #2: Seems that it was my secondary screen the one that didn't show the color! Tried with another screen and I could see the progress clearer... perhaps a stronger color will make it easier to see across different devices?  Update: Seems to be an issue in Firefox, as I tried on Chrome and the progress is visible.    I just noticed that the badge progress on your user profile had it's style changed (don't know when but it seems that recently).  However, seems that the progress of gold badges is not working properly. If we see the Bronze and Silver badges, they are "filled with color" as you progress on the badge, however golden ones are not colored (just the border).   Here a picture illustrating what I say:    Is there a reason for this behavior? Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line. Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line.   I am trying to argue from the definition of uniform convergence for a sequence of real-valued functions, but am struggling quite a lot. My efforts so far have concentrated on trying to find a sequences, ${a_n}$ which tends to zero, such that   $$|(1+\frac{x}{n})^n -e^x |\leq a_n$$ for all $n$. But I have been unsuccessful thus far. All help is greatly appreciated. I remember reading this book in the late 80's or early 90's. It was a - sort of - semi SciFi contemporary fairy tale.  It was set in Manhattan in contemporary times, near future and (I think) late 1800's. There was a mist wall round New York that ships could disappear into and arrive out at different times.  There was a man with a white horse, the horse could fly but had forgotten how. The man is brought forward from the past but (like the horse) forgets who he is.  An engineer arrives from the future in a gigantic ocean liner and builds a bridge of light to heaven (possibly at the last millennium turn over).   It was a very involved book with a huge cast of characters that all had some part to play in the story.  In the 1800's story there was a girl suffering from consumption, she slept in a glass room on the roof of her house, open to the freezing winter air, it was what kept her alive. She was rich (the family had made money in whaling) but somehow she and The Man (the guy with the white horse, who was pretty much a nobody) met and fell in love (I think).  Some of the story was set around a lake in upstate NY, and one character started off in San Francisco (contemporary times) and traveled to the lake.  Anyone any idea what I'm talking about? Print lines between (and excluding) two patterns I'm going to submit form using cURL, where some of the contents is come from other file, selected using sed  If param1 is line matching pattern from other file using sed, below command will works fine:  curl -d param1="$(sed -n '/matchpattern/p' file.txt)" -d param2=value2 http://example.com/submit   Now, go to the problem. I want show only text between 2 matching pattern excluding the matching pattern itself.  Lets say file.txt contains:  Bla bla bla firstmatch It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. secondmatch The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English.   Currently, lots of "beetween 2 matching pattern" sed command won't remove firstmatch and secondmatch.  I want the result to become:  It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. I do not want packages to be installed under /. (sudo apt-get install ...).   The reason why I need to do this is that:  My '/' is mounted with my SSD(16GB) and my hard disk(500GB) is mounted at '/home'.   I have already installed my Ubuntu 12.04 on that SSD. There is not much free space under '/' now. Let $f: \mathbb R^2 \to \mathbb R$ defined as   $f(x,y)=(x^2+y^2)\sin(\dfrac{1}{\sqrt{x^2+y^2}})$ if $(x,y) \neq (0,0)$, $f(x,y)=(0,0)$ if $(x,y)=(0,0)$  Prove that $f$ is differentiable at $(0,0)$ but that the partial derivatives of $f$ are not continuous at the origin.  The attempt at a solution:  I had some problems at the first part of the exercise:  I've calculated $\dfrac{\partial f}{\partial x}(0,0)$ and $\dfrac{\partial f}{\partial y}(0,0)$, both exist and are equal to $1$.  So, to prove $f$ is differentiable at the origin, I have to show that   $\lim_{(x,y) \to (0,0)} \dfrac{|f(x,y)-(f(0,0)+&lt;\nabla f(0,0),(x,y)&gt;)|}{\|(x,y)\|}=0$ But $\lim_{(x,y) \to (0,0)} \dfrac{|f(x,y)-(f(0,0)+&lt;\nabla f(0,0),(x,y)&gt;)|}{\|(x,y)\|}$ equals  $\lim_{(x,y) \to (0,0)} \dfrac{|(x^2+y^2)\sin(\dfrac{1}{\sqrt{x^2+y^2}})-(x+y)|}{\|(x,y)\|}$. I don't know how to show this limit equals $0$.  For the second part (I've only analyzed $\dfrac{\partial f}{\partial x}$ since both cases are the same), what I did was try to show that $\lim_{(x,y) \to (0,0)} \dfrac{\partial f}{\partial x}(x,y) \neq \dfrac{\partial f}{\partial x}(0,0)$   I thought of considering the curve $\phi(t)=(t,0)$, so   $\lim_{t \to 0} (f\circ \phi)(t)=\lim_{t \to 0} 2t\sin(|t|^{-1})-\dfrac{t\cos(|t|^{-1})}{|t|^{-1}}$. But $\lim_{t \to 0^{+}} \dfrac{t\cos(|t|^{-1})}{|t|^{-1}}=\lim_{t \to 0^{+}} \cos(\dfrac{1}{t})$ and this limit doesn't exist, so the original limit doesn't exist. This shows the partial derivative with respect to $x$ at the point $(0,0)$ is not continuous.   I would appreciate if someone could help me to finish the first part and tell me if what I did in the second part of the problem is correct. Why does the C++ map type argument require an empty constructor when using []? Why can some holodeck matter leave the holodeck, while other holodeck matter (notably people) cannot? Proof of dividing fractional expressions For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves  $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}?$$ Cannot log in with Apple ID after performing password reset My wife forgot the password to her Apple ID account. Her Macbook Air (OS X Yosemite) restarted to install some updates. She has her user account set up through her Apple ID account.  Since she forgot the password she reset it from my laptop.  She can log in to iCloud and Apple ID with the new password, but cannot log in with the new password on the Air.  When she tries to log in she is greeted with this message:     "This account uses an iCloud password to log in. If you need to change   the password, use another device to change it at appleid.apple.com." Let he who believes in this prophet speak now what he knows.       Let him who believes in this prophet speak now what he knows. RAII and smart pointers in C++ In practice with C++, what is , what are , how are these implemented in a program and what are the benefits of using RAII with smart pointers? Can we get our personal voting data displayed on the Questions lists? There's been several times where I have been surfing the site, and noticed a question that I am not sure I've checked. Even if I've checked the question before, I can't recall off-hand whether or not I've voted on it or its answer.  Is there a way we can have our personal voting history reflected to some degree or another, in the Questions listing? For example, could there be an indicator as to whether or not we have already voted (up or down) on a question, and perhaps maybe an additional indicator as to whether we've up-voted any of its answers?  It would be very convenient to have some of this data available up-front, without actually having to view the question again. What are all the correct answers to the Squirrel's questions, and what is the reward when I get them all correct? Using oscilloscope safely with AC mains I know there is quite few discussions already on this topic, but since English isn't my first language, I though it better to get some more information about this.  I have used oscilloscopes to troubleshoot electronics problems and measure AC engine current spikes at startup, but I have no experience using them to measure high voltage levels and my experience is with old analog scopes.  So now I have this new DSO and the input rating is 400V peak to peak AC, so it can handle about 140V AC RMS? Not that I'm going to input that much there.  Since I live in Europe the mains here are 230 V @ 50 Hz I can't measure them directly.  If I set the attenuation from the scope and probe to 10X, the voltage is 23 RMS and it is safe to measure? And yes the probes are rated for 1kV.  If that is true, it's also safe to measure voltage between two main lines (400V RMS), since the peak-to-peak would be about 112 volts?  SAFETY: When I am measuring AC, I will be using the scope from its internal battery or with a isolating transformer (1:1) so that it is floating, meaning that there is no connection to earth ground. Are there any other safety issues I have not realized? Why do the measurements of this object seem erroneous? Is there an easy way to convert a number to a word in PHP? In PHP, is there an easy way to convert a number to a word? For instance, 27 to twenty-seven. Possible orders of permutations of $11$ symbols