I would like to show that $\mathbb{Z}[\phi]$ is a UFD, where $\phi^2+\phi+1=0$. I started with a division with remainder to determine the irreducibles, but I could not finish that, and I do not know how to approach this problem. Could you provide me with some resources or show how I can solve this? Thanks in advance.
Is $\mathcal{O}_{\mathbb{Q}(\sqrt{5})} = \mathbb{Z}[\phi]$, where $\phi={1+\sqrt{5}\over 2}$ is the golden ratio?  I know that $5 \equiv 1 \mod 4$, so that then $\mathbb{Z}[\sqrt{5}]$ is not closed as far as integers go. But I'm a little confused cause  $\mathcal{O}_{\mathbb{Q}(\sqrt{-3})} = \mathbb{Z}\left[-\frac{1}{2} + \frac{\sqrt{-3}}{2}\right]$? So how do I know when to use $\frac{1}{2}$ rather than $-\frac{1}{2}$? Which one applies for $\mathcal{O}_{\mathbb{Q}(\sqrt{5})}$?
Proof of the uncountability of reals using the diagonal argument—problem? Title tag different from title appearing in Google? I'm just wondering how it is possible that IGN.com has this in its title tag:      Video Games, Cheats, Walkthroughs, Game Trailers, Reviews, News, Previews &amp; Videos at IGN   while on Google its title is simply "IGN".   How can one achieve such a thing? Does it use some special meta tag I can't see? Nature of the quantum degeneracy pressure It is commonly known that it is the electron degeneracy pressure that prevents the collapse of a white dwarf into a neutron star, and it is not the electromagnetic force. However, it is also widely accepted that all the forces in the universe are just manifestations of the four elementary forces in the universe. So can the degeneracy pressure, caused due to Pauli's Exclusion principle, be shown to be due to one of these forces? I have read some other questions in SE , , and they do not answer my question fully.  In one of the links, the degeneracy pressure is described as an 'entropic force', but as far as I understand, all entropic forces can be explained by the underlying fundamental forces. If the degeneracy pressure is something that is fundamentally different from the other forces, why isn't it classified as a separate force? Integrals of Pullbacks How do they decide Blu-ray/DVD release date? I think my question is different from .  Some movies' productions release Blu-ray/DVD so early after theatrical release, others will take time. How do they decide for the Blu-ray/DVD release date? How to modify an invalid '/etc/sudoers' file? I would like to create a minecraft server for me and and my boyfriend to play on.  How do I set it up? Do I need any additional software? Do you have some links to walkthroughs or tutorials?  I have tried to set this up a few times but have failed every time. How can the speed of sound increase with an increase in temperature? I am not sure what caused this error, but here is what the whole error says, and also this is sitting as a notification, and preventing me from updating any software using Update Manager - Please provide some assistance or tell me how to figure out what to do to fix it.  Could not calculate the upgrade  An unresolvable problem occurred while calculating the upgrade.  Please report this bug against the 'update-manager' package and include the following error message: 'E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.' Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$ How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$ Mathematical operators, such as function names, should be set in roman type, not italics. Latex already has commands for some operators, including \max, \min, and \log.  How can I define additional such commands? Not enough space in /var/cache/apt/archives/ Why is there not a simpler way to calculate the standard deviation? Steps of getting standard deviation. :        Work out the average (mean value) of your set of numbers   Work out the difference between each number and the mean   Square the differences   Add up the square of all the differences   Divide this by one less than the number of numbers in your set -   this is called the variance   Take the square root of the variance and you've got the standard   deviation      Am I missing out something, or why do we need to square the differences in step 3?  Why not simply do a Abs (multiply all negative numbers by -1) in step 3?  Also, my second question is why do we need to divide by one less than the number of numbers in the set in step 5? why not simply divide by the number of numbers? How to get ongoing notifications of unread texts? One of my family members texts me at night when they are having nightmares. I want to know when they text me at 4 am instead of sleeping through it.  Is there a way to make text notifications keep going off until read? Positive correlation and negative regressor coefficient sign Is it possible to obtain a positive correlation between a regressor and a response (+0,43) and, after that, obtain a negative coefficient in the fitted regression model for this regressor?  I'm not talking about changes in the sign of the regressor among some models. The coefficient sign always remains.  Could  the remaining variables of the fitted model influence the changing of the sign? How do I autohide the top panel in Unity? When windows are not maximized, the top panel is basically wasted space for me in Unity.  Is there a way to make it autohide? i.e. make it appear on mouse over. Code   find / -name netcdf   Output  find: `/root/.dbus': Permission denied find: `/root/.gconf': Permission denied find: `/root/.gconfd': Permission denied find: `/root/.gnome': Permission denied find: `/root/.gnome2': Permission denied find: `/root/.gnome2_private': Permission denied permanent hat-like thing? I really like how, in the winter bash, there are hats. They are pretty cool and actually encourage me to get badges, something I didn't care about before. I think there should be something like this, but all year round, to encourage people to get badges. Badges, to me, seem currently pretty useless and I never want to get any because...there is no good reason to. Hats: well, hats are just awesome Playing a video in full screen after startup window.onbeforeunload and window.onunload is not working in Firefox, Safari, Opera?