Not sure how to do this one. If $S$ is a field, then I was considering that $\exists r_1,\ldots, r_n\in R$ s.t. $1 = r_1s_1+\cdots+r_ns_n$ so for $r = rr_1s_1+\cdots+rr_ns_n$. Maybe that is somehow useful for taking inverses of elements.    The assumption that $S$ is an integral domain is necessary because otherwise we could have $S = \mathbb{Z}_p[x]/f(x)$ where $f(x)$ is not irreducible. This is still a finitely generated $\mathbb{Z}_p$-module, but its not a field.   Any hints or solutions would be much appreciated. I feel like this isn't that hard and I'm missing something simple  Source:
I'm reading Galois Theory by Steven H. Weintraub (second edition), and finding that I'm at least somewhat short on the prerequisites. However the following proof looks wrong to me - am I misunderstanding something, or is it actually an incorrect proof?  Lemma 2.2.3. Let $F$ be a field and $R$ an integral domain that is a finite-dimensional $F$-vector space. Then $R$ is a field. Proof. We need to show that any nonzero $r \in R$ has an inverse. Consider $\{1, r, r^2, \cdots\}$. This is an infinite set of elements of $R$, and by hypothesis $R$ is finite dimensional as an $F$-vector space, so this set is linearly dependent. Hence $\sum_{i=0}^n{c_i r^i} = 0$ for some $n$ and some $c_i \in F$ not all zero.  It then goes on to show, given the above, that we can derive an inverse for $r$. However, if I consider examples like $r = 2 \in Q[\sqrt{2}]$, $r = \sqrt{2} \in Q[\sqrt{2}]$ or $r = 2 \in Q[X]/{&lt;X^2&gt;}$, the set $\{1, r, r^2, ...\}$ doesn't look linearly dependent to me. I do believe the lemma is true (and might even be able to prove it), but this does not look like a correct proof to me. Am I missing something? [Edit] Well yes, I am. Somehow I had managed to discount the possibility of any $c_i$ being negative, despite repeatedly looking at each fragment of the quoted text in an attempt to find what I might be misunderstanding.
Is there a tool that combines zcat and cat transparently? When handling log files, some end up as gzipped files thanks to logrotate and others not. So when you try something like this:  $ zcat *   you end up with a command line like zcat xyz.log xyz.log.1 xyz.log.2.gz xyz.log.3.gz and then with:  gzip: xyz.log: not in gzip format   Is there a tool that will take the magic bytes, similar to how file works, and use zcat or cat depending on the outcome so that I can pipe the output to grep for example?  NB: I know I can script it, but I am asking whether there is a tool out there already. Performing deep learning land cover classification using R? I am really new to Deep Learning and, unfortunately, I can't find example codes on land cover classification other than  where the author wrote a script in R for a large dataset.   The main reason that I am asking is because recently I found a few papers on Remote Sensing Image classification using Deep Learning and I was wondering if there were any R examples on that subject.  So, is there a code example, preferably a step-by-step, which I can use?  I found several git repos like  that can do Deep Learning classification with R, BUTT, I am still looking for a simple multi-spectral image segmentation method. An example of that can be found at  and I also found a similar question in  for which the answer is "use something else other than R if you can".  If I find anything that works today I will be posting a working answer. Did they decide to destroy it simply because they were Wise and knew it was dangerous to wield? Did they create the Fellowship simply because they were heroic; and decided to face Sauron after the Ring was destroyed?  Or did they know that destroying the ring would defeat Sauron, and they wouldn't actually have to do anything once that is taken care of? Dimensioning of a technical drawing in TikZ Is there is an easy way to mark dimensions in a technical drawing with TikZ?    Is there a library or something?  Edit  I am using XeLaTeX.  Update  I chose Martin's answer because it serves me well for the moment. Ultimately the best solution would be a library that, in an easy and consistent manner, would allow to change the arrow/dimension lines styles, would support polar coordinates, would allow to choose 2 nodes and the vertical distance that you want the dimension to be typeseted etc. I have an expression for a covariance matrix $C$ in terms of the indices $i$ and $j$.  In this way I can analytically calculate the elements of my covariance matrix, however when I try to invert $C$ matlab gives a warning about the matrix being close to singular.  The inversion therefore doesn't work, by which I mean the multiplying $C$ by the inverse given i do not get the identity.  I have tried calculating the pseudo inverse but this also does not work.  I have also tried adding a small constant along the diagonal but again the results do not work.  In general I'm working with matrices of dimension 1200 but I will give low dimensional example matrix that has the same properties, i.e. the matrices are symmetric about the diagonal and the anti-diagonal     19.9939  19.9954  19.9958  19.9951  19.9933  19.9905  19.9865    19.9954  19.9973  19.9981  19.9978  19.9965  19.9940  19.9905    19.9958  19.9981  19.9993  19.9995  19.9985  19.9965  19.9933    19.9951  19.9978  19.9995  20.0000  19.9995  19.9978  19.9951    19.9933  19.9965  19.9985  19.9995  19.9993  19.9981  19.9958    19.9905  19.9940  19.9965  19.9978  19.9981  19.9973  19.9954    19.9865  19.9905  19.9933  19.9951  19.9958  19.9954  19.9939   As mentioned in the title the matrix isn't positive, however the the negative eigenvalues are very small suggesting that the matrix is not positive only due to machine precision.  The negative eigenvalues are $-1.4048e-14$ and $-2.4571e-15$.  How can I go about inverting these matrices? Facebook: Ways to preserve image quality of uploaded images? After uploading an image, Facebook converts it to a new image.  Their conversion process introduces artifacts and mottling to the magenta-ish parts of the image.  Original image:   Facebook's conversion:   Are there ways to preserve the original quality of this image and have the image appearance on Facebook reflect more of the original quality? Derivative of an even function is odd and vice versa When should I use rich black? Windows seems to have a length limit on file names when trying to delete, though it won't prevent those files from being created.  Our build process creates a number of temporary files (many build off of a WSDL) that run afoul of this limit.  Our ant script is somehow able to delete them when doing a clean, but sometimes I need to delete the workarea directory (where all the temp files go) without actually doing a full clean from ant.  This is the same errors , but the answers there don't really work for me as I'm dealing with a directory, not a file, and I don't always know what specific files or subdirectories are causing the problem.  And I'm trying to avoid any manual process (other than triggering a single command) to actually delete them.  If I try deleting the directory from Explorer I get the error   Cannot delete [file name]: The file name you specified is not valid or too long.   Specify a different file name   Trying Remove-Item in powershell gives the following error:  Remove-Item : The specified path, file name, or both are too long. The fully qualified file name must be less than 260 characters, and the directory name must be less than 248 characters. At line:1 char:12 + Remove-Item  &lt;&lt;&lt;&lt; -force -Recurse &lt;directory&gt;   Does anyone know of any tools or easy ways to get around this delete error without having to manually find the problem files and move/rename them? Overview:  Lots of games with RPG-like statistics allow for character "buffs", ranging from simple "Deal 25% extra damage" to more complicated things like "Deal 15 damage back to attackers when hit."  The specifics of each type of buff aren't really relevant.  I'm looking for a (presumably object-oriented) way to handle arbitrary buffs.  Details:  In my particular case, I have multiple characters in a turn-based battle environment, so I envisioned buffs being tied to events like "OnTurnStart", "OnReceiveDamage", etc.  Perhaps each buff is a subclass of a main Buff abstract class, where only the relevant events are overloaded.  Then each character could have a vector of buffs currently applied.  Does this solution make sense?  I can certainly see dozens of event types being necessary, it feels like making a new subclass for each buff is overkill, and it doesn't seem to allow for any buff "interactions".  That is, if I wanted to implement a cap on damage boosts so that even if you had 10 different buffs which all give 25% extra damage, you only do 100% extra instead of 250% extra.  And there's more complicated situations that ideally I could control.  I'm sure everyone can come up with examples of how more sophisticated buffs can potentially interact with each other in a way that as a game developer I may not want.  As a relatively inexperienced C++ programmer (I generally have used C in embedded systems), I feel like my solution is simplistic and probably doesn't take full advantage of the object-oriented language.  Thoughts?  Has anyone here designed a fairly robust buff system before?  Edit: Regarding Answer(s):  I selected an answer primarily based on good detail and a solid answer to the question I asked, but reading the responses gave me some more insight.  Perhaps unsurprisingly, the different systems or tweaked systems seem to apply better to certain situations.  What system works best for my game will depend on the types, variance, and number of buffs I intend to be able to apply.  For a game like Diablo 3 (mentioned below), where nearly any bit of equipment can change a buff's strength, the buffs are just character stats system seems like a good idea whenever possible.  For the turn-based situation I'm in, the event-based approach may be more suitable.  In any case, I'm still hoping someone comes along with a fancy "OO" magic bullet which will allow for me to apply a +2 move distance per turn buff, a deal 50% of damage taken back to the attacker buff, and a automatically teleport to a nearby tile when attacked from 3 or more tiles away buff in a single system without turning a +5 strength buff into its own subclass.  I think the closest thing is the answer I marked, but the floor is still open.  Thanks to everyone for the input. Can domain masking be used to move content from a subdomain to a subdirectory? example.com is hosted on IP address c.c.c.c at host1, no access any windows server admin at all blog.example.com is  on IP a.a.a.a at host2, full access to Linux server admin, using WordPress   So basically, I can fiddle with subdomain all I want.  I can't 301 redirect because the content on host 1 is an eCommerce site that I have no server control over. Can't put WordPress on it, can't run scripts, can't do anything on the server period.  Is there a method to "mask" all the URLs on root blog.example.com so they appear as example.com/blog/ folder?  They can't actually be sent to example.com/blog/ as that doesn't exist. It just needs to look to the surfer as if they are there.  Can this be done with some kind of masking or something?  Will this be better for SEO as hopefully search engines will see the blog in a directory on the main domain? In the series finale, Admiral Janeway travels back to 2378 to take Voyager back to Earth. Out-of-universe, of course, this point in time was chosen to fit in with the continuity of the rest of the series. In-universe, however, it seems that Janeway could have chosen any point in time, since she had the ability to travel to and from the Delta Quadrant at will. Is there a reason given, in-universe, for why she chose that particular incident to travel back to? In Harry Potter and the Philosopher's Stone, Hermione is in the toilets when Harry and Ron lock in the troll. She lies and says that she went looking for the troll. Why? She could have just said that she was in there because she needed the toilet (she was allowed to be, it's a girls' bathroom) and Harry and Ron came to save her.   Is there a reason she lied? Here is the problem:  Suppose that $*$ is an associative binary operation on a set $S$. Let  $$H:= \{a \in  S\mid a * x = x * a \mbox{ for all }x\in S\}.$$  In other words, $H$ is consisting of all the elements of $S$ that commute with every element in $S$.  Show that $H$ is closed under $*$. Eclipse: How to build an executable jar with external jar? I am trying to build an executable jar program which depends on external jar downloaded. In my project, I included them in the build path and can be run and debug within eclipse.  When I tried to export it to a jar, I can run the program but I can't when I try to press a button which includes function calls and classes from the external jar. I have edited the environment variables (Windows XP) CLASSPATH to include paths of all the external jar, but it doesn't work.  A point to note is that I got compile warnings while exporting my executable jar, but it doesn't show up any description about the warnings.  Would someone kindly provide a thorough guide on how to include an external jar program using eclipse? Show tabs in Chrome full screen mode? Firefox can do it, can Chrome?   Here's what I'm looking for: Plot an elliptic curve in Latex I have filed my 2012 tax return jointly with my wife, along with my wife's ITIN application, back in late March.  In April, I got a letter for IRS ITIN department saying that we should send my wife's original passport to them. So we sent my wife's passport on May 15th. USPS tracking number confirmed that IRS received it on May 20th.  However, last week (end of June) we have received another letter from IRS ITIN department saying that our ITIN application has been rejected, rejected because we have not sent the original identification document that they requested!  Okay, let's put our tax return and ITIN aside. Where is my wife's passport? Right now my wife does not have her personal identification, nor proof of legal status in U.S. Could IRS have lost it?  I tried to call the IRS several times. One time I waited for 1 hour and 20 minutes to talk to a person, and that person said he has only 10 minutes to talk before he has to leave. He said they'll investigate and someone(?) will call us back soon, but never called back. I wrote them a letter, but no response. I tried calling several times more, but machine is saying 'sorry too many calls right now, we can't accept your call'.  What can I do? Word for two collections that do not have any elements in common I'm looking for a word for when you have a collection A and a collection B and they have no overlap.  In mathematical terms: the relation between two sets where the intersection is empty. Like in this Venn diagram:    The word that keeps popping up in my mind is disparate sets, but I don't feel that completely covers the meaning when I look at the definition, because that doesn't seem to preclude overlapping sets that are different:     disparate   1. distinct in kind; essentially different; dissimilar:   disparate ideas. How do I challenge an audit failure? I've run through Both SO and Meta searches for an answer, as well as reviewing the , and I cannot find a canonical process to challenge an audit failure.  I can find discussions of the single-choice "I Understand" topic that appear unresolved, in that no solution presently exists.  The particular question in point is .  Despite the number of up-votes, I feel that this quite clearly falls into a common closure category: "Here's my problem; how do I write the program?"  Compare this to   I also failed review on the first link I posted above, .  This looks like a case of code review, rather than SO.  I don't want to get rid of audits; I've learned good things from some of my failures, especially at the start.  However, in the past few weeks, a majority of my failures have been highly questionable, and I've drawn more than one suspension from those that are outright wrong by current standards.  If we have a challenge mechanism, please point me in the right direction.  Otherwise, it's simply not worth the annoyance any more, and I'll drop back to merely comments and answers.