Add a "Did you mean" feature for tags
What's the possibility of adding a "Did you mean ..." in the tagging system?
In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?  Using the Wolfram Alpha site, this input gave an almost-square: PolarPlot(0.75 + ArcSin(Sin(2x+Pi/2))/(Sin(2x+Pi/2)*(Pi/4))) (x from 0 to 2Pi)  This input gave an almost-octagon: PolarPlot(0.75 + ArcSin(Sin(4x+Pi/2))/(Sin(4x+Pi/2)*Pi^2)) (x from 0 to 2Pi)  The idea is that as the number of sides in a regular polygon goes to infinity, the regular polygon approaches a circle. Since a circle can be described by an equation, can a regular polygon be described by one too? For our purposes, this is a regular convex polygon (triangle, square, pentagon, hexagon and so on).  It can be assumed that the centre of the regular polygon is at the origin $(0,0)$, and the radius is $1$ unit.  If there's no such equation, can the non-existence be proven? If there are equations, but only for certain polygons (for example, only for $n &lt; 7$ or something), can those equations be provided? Why is the following C# code not allowed:  public abstract class BaseClass {     public abstract int Bar { get;} }  public class ConcreteClass : BaseClass {     public override int Bar     {         get { return 0; }         set {}     } }      CS0546 'ConcreteClass.Bar.set': cannot override because 'BaseClass.Bar' does not have an overridable set accessor how to pass a value to a variable that in a other variable I am getting a "bad substitution" error while running this code. I want to print the values in h_1, h_2, h_3with a for loop.  #/!/bin/bash h_1=12 h_2=13 h_3=14 for ((i=1; i&lt;=2; i++))  do   echo "${h_$i}" done I am really having trouble working a problem in my course. I will post it and then add what I have thought etc.  Let p be an odd prime, then using the Euclidian Algorithm, calculated the GCD of of $x^{p-1}-1$ and $x^2+1$ in $\mathbb{Z} / p\mathbb{Z}[x]$  Also, how does $p=1 (mod4)$ tell us that  $-1$ is a square in $\mathbb{Z} / p\mathbb{Z}$, infact that $-1$ is a square here if and only if $p=1(mod4)$.  My thoughts:   I know that $$x^{p}-x=(x)(x-1)(x-2)…(x-(p-1))$$ and so that in this sense our roots will be the congruence classes of $1,2,…p-1$  and also that $$x^{p-1}-1=(x-1)(x-2)…(x-(p-1))$$  But I get lost when trying to apply the EA to these polynomials,  like how can I do it when p is arbitrary?  Preferably id like to be able to understand/see how to do the first part first i.e. using division with residue and then move on to the second part. Any insight/hint or solution to this would be great,  Thank you all  Update:  When I try to do division with residue, I just get  $$x^{p-1}-1=(x^2+1)(x^{p-3}-x^{p-5}+x^{p-7}-x^{p-9}...)$$  for example,  $p=7$  $$(x^{6}-1)=(x^2+1)(x^4-x^2+1)$$  for $p=11$  $$(x^{10}-1)=(x^2+1)(x^{8}-x^{6}+x^{4}-x^{2}+1)$$  which for specific cases tells me that 1 is the gcd, but I dont know how to generalize that for the case with arbitrary prime p. Helping someone climb I know this is kinda stupid but what is the word for "helping someone climb by allowing him to stand on your hands" (to allow him/her see above some wall or climb something)? I am a Saint Lucian and will be traveling to Manchester, UK soon about which I am a bit nervous.  Run Down:   I own a small business in my country (in the Caribbean), I have a certificate mentioning the details of the business from my Government. I pay rent in my country, but I do not have any loans (I hate owing). I pay life insurance in my country. I have a US B-1/B-2 visa (Visitor &amp; Business). I have visited the US once (last year November) on my US B-1/B-2 visa.   What I will be doing in the UK:   I plan to spend three months out of the six months which would be granted if I get leave to enter stamp. I plan to skydive in the first week and spend a few nights at an Airbnb. I plan to visit the Manchester Football Club stadium, I'm not the biggest of football fans but since I'm in Manchester, why not! I plan to do some more sightseeing and enjoy my time there. After the first week I will be staying with my female friend who owns an apartment for the rest of the time showing me around and helping me experience the UK lifestyle until June.   Why I'm leaving in June:   I am a WordPress enthusiast (I'm a Programmer) and WordCamp EU will be happening in France this year so I plan to fly from the UK to Paris where I will spend three days then take another flight from there to my country.   How my flights will be:   I will be taking a one-way Thomas Cook flight from my country to Manchester but will be purchasing my ticket for Manchester to France ahead of time so I could show Thomas Cook as well as Immigration that I don't plan to stay.   My concerns, could someone address them point by point please?:   I have a bank account but I just use it for collecting easy payments from my clients and pay for expenses. I store my savings in a credit union because of higher interest rates paid than the bank. When I applied for my US visa I showed a credit union statement showing my total balance. I'm seeing online that I should provide statements but these statements would look very bad since I my bank account is just for expenses... I mean it sometimes gets as low as $30 then, when my monthly earnings come in from clients, I withdraw savings to the credit union and leave what I need in the account for expenses. I read another post and it said people with bank statements showing poor spending might have difficulty but it's simply because the account is used for expenses not savings. For the later stages of my visit (after the first week) I will be staying with my friend. I will simply be staying with her to save on cost of Airbnbs for the next two months three weeks, even though they could be cheap, free is better right? I will be booking the Airbnb, skydiving and tours prior to traveling to the UK, will that help? Is my stay too long for staying with someone? I won't be depending on her for anything since I have my own funds...it's just convenient that she has a place where I could stay. If it is then I don't have a problem shortening my stay to 1-2 weeks and just head back to my country and take a separate flight to Paris in June.   I am nervous because I really want to visit the UK and see my friend, I am also not sure if the minimum amount of time that could be given to a visitor is six months? What if I get a round trip for two weeks and the Immigration officer gives me exactly two weeks stay, is that possible? Or is the minimum they could give six months?  Your help is very much appreciated, I want soon to try backpacking (nomad life) but I don't think my poor heart could take the stress of worrying about immigration :(  Update  Since people in the comments are asking about my relationship with the friend I plan to stay with even though she's from my country originally but now a UK citizens and we are simply friends. What are the most suitable filter holders for my Samyang 14mm autofocus prime lens? I've looked into the ones from Haida and Nisi which are both suitable, though they're quite pricey for me currently - does anybody know of any cheaper possible alternatives? I understand there's the 160mm specifically made for use with this lens, though obtaining filters for it is difficult and limited. I'm hoping for something which may be cheaper than both Haida and Nisi, and still suitable for the 14mm lens. Thanks! CK2 is a big game with lots of variables, so I'll stick to adding them upon being requested to minimize the hassle.  Anyway, let's say that you have a lot of children and the oldest one claims all the land according to laws of succession. After the father dies, the circumstances will obviously piss off a lot of people (in my case that would be 6 out of 8 vassals and 4 siblings with an opinion of -50 and below one of which is a sister married to the Basileus of the Byzantine friggin' Empire, a case of excommunication, 4 failed assassination attempts under 3 years... and lots and lots of vikings). Let's also say that the current playable character is completely incompetent compare to the next sibling in line.  So, what would be the most optimal methods to off myself without further screwing over the countryside so the next heir in line has to fix it (if we assume that said heir is actually capable of doing so)?  This is my list of DLCs (not including music and unit packs):   Charlemange Legacy of Rome Sword of Islam The Republic Way of Life I am using lightning:recordEditForm and lightning:inputField for creating a new record. this component is being used inside a lightning community builder. When i click the submit button, i am able to create a new record in my custom object. However i need the created record ID as i need to pass it in the URL for the next page. What blueprints become available during story? As I progress, I realize that some blueprints open up throughout the story and don't have to be researched. Are these random or are they always the same and is there a list over those? Seems you can easily waste research point if you don't know that. hard disks spin up by processes / applications that simply get a list of disks? How to prevent? I have a 24/7 always-on Debian Jessie based headless home server that has a large 1TB SSD for the OS and all of my frequently accessed files. This same system has 4 larger hard disk drives in a SnapRAID array. These are mainly for archiving infrequently accessed Blu-rays and want those drives to remain spun down in standby unless I actually read or write to them. They are all formatted as ext4 and mounted with noatime and nodiratime enabled.  So even though no process or program should be regularly accessing those drives in any direct way, the hard drives constantly get spun up from standby. It seems to be related to graphical programs that provide a gui file browser, even something like Chromium. If I don't even browse into those drives, I'm thinking that these processes by simply getting a list of available drives spins up the hard disks. Much like blkid does. The problem is, it's hard to determine the root cause of this since none of these processes are actually reading or writing the filesystem on those drives, so no files are actually changing or being touched. Is there some sort of cache that I can populate or a buffer to prevent these programs from spinning up the hard drive simply by getting a list of available disks? This is honestly driving me insane, since I can't find a reliably way to keep these disks spun-down even though there is no direct access of the filesystem.  UPDATE: Thanks to Stephen's answer, I was able to trace the disk activity to gvfs and udisks. It's a real shame that these processes insist on waking up disks in standby when they aren't actually being accessed to do any real I/O with the filesystem. So far I just uninstalled them, knowing that it will remove some functionality from PCManFM and the like. Is it possible to generate electric energy from the radiation of radioactive materials I wonder if it is possible to generate electric energy from the radiation of radioactive materials like nuclear waste? If it is then wouldn't that also mean that it could be used as an energy source for a very long time since the radiation takes a long time to decay? Can there be a notification for those people that participated in the question in the past (either who asked or answered it) once a bounty is started on it? Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $ Inspired by , I was wondering if there is a simple logical argument to   Show that $ a,b, \sqrt{a}+ \sqrt{b} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b} \in\mathbb Q $  Note that the original link is using a computational method, where as I am looking for a simple logical argument.  I tried (unjutifiably) to argue that if some of two square roots is rational then each one is rational, this is a different than the (incorrect) argument that if sum of two algebraic numbers is rational then each one is rational ( counter example $a=1-\sqrt{2},b= 1+\sqrt{2} $) How do I remove an OpenID or login? How do I remove login credentials (emails or OpenIDs) from my account?     For more information, see "" in the . Can Ubuntu 32-bit access 8GB RAM? I need to use a large software package (Xojo) that requires the 32-bit architecture to run. Try as I might, I can't seem to install the 32-bit libraries on my 64-bit installation of 15.04.  I don't have a problem running a 32-bit version of Ubuntu (I'm only using this computer for Xojo) but the computer has 8GB of RAM and I'd like to be able to use it all. Is there a way to utilise all 8GB of RAM with a 32-bit Ubuntu install? Does Ubuntu do this out of the box? Where is the BIOS stored? From :     BIOS software is stored on a non-volatile ROM chip on the motherboard. ... In modern computer systems, the BIOS contents are stored on a flash memory chip so that the contents can be rewritten without removing the chip from the motherboard. This allows BIOS software to be easily upgraded to add new features or fix bugs, but can make the computer vulnerable to BIOS rootkits.   Since ROM means Read-Only Memory,  why can the BIOS contents be rewritten?  Does the "flash memory chip"  mean the same as the "non-volatile ROM", both meaning where BIOS is stored? What do Pentax and Sigma DSLRs offer that differs from Canon and Nikon? We have a question and answers outlining the . I'm interested in the other DSLR makers, and how the cameras they make compare in terms of significant features, system design philosophy, and unique or interesting photographic capabilities enabled or made easier by these cameras. Likewise, what such things are not handled as well?  Take "the big two have more market share" and the associated advantages of availability, accessibility, and third-party support and documentation .  I'm not interested here in compact cameras, interchangeable-lens cameras with electronic viewfinders, or in rangefinders. (If you're interested in that, though, see )  P.S.: Differences in lens lineups are covered at If nonempty, nonsingleton $Y$ is a proper convex subset of a simply ordered set $X$, then $Y$ is ray or interval? This is from Exercise 7 in p. 92 in Munkres's Topology.  Except for the trivial cases such as $Y$ is empty set or singleton, it seems if $Y$ is convex in an simply ordered set $X$ then $Y$ is interval or ray.  But I cannot start my proof because I cannot use $\sup$ or $\max$ functions.  What should I do?