This problem once again is from a previous exam.  The problem is to compute the group of automorphisms of the ring $S$ where $S=\mathbb{Q}[x]/(x^2)$  My thoughts: Well $x^2$ is reducible over $\mathbb{Q}$ so I am completely stuck. Also what exactly is the theory I need to know to solve this problem?. I don't even know where to look in the text book to find the relevant theory for this. In particular  I have not learned Galois Theory. In fact we don't need to know Galois' theory for my exam. So I'm hoping this doesn't use any of that heavy machinery.  Can you guys help?  Thanks for your help and answers.
Here's the problem: if $S = \mathbb{Q}[x]/(x^2)$, compute the group  of automorphisms $\mbox{Aut}(S)$.   First off, I have the solution and I know it's isomorphic to the multiplicative group $\mathbb{Q}\backslash\{0\}$, so I'm trying to understand it. It uses what is called the substitution principle and the fact that the restriction of any endomorphism $\phi: \mathbb{Q}[x]\to \mathbb{Q}[x]$ to $\mathbb{Q}$ is the identity. It says that these two facts implies that the endomorphisms of $\mathbb{Q}[x]$ are in 1-1 correspondence with $h \in \mathbb{Q}[x]$, where the endomorphism corresponding to $h$ is $\phi_h : \mathbb{Q}[x] \to \mathbb{Q}[x]$ given by $\phi_h(f) = f(h)$ for $f \in \mathbb{Q}[x]$. I guess I just don't understand how it applied the substitution principle? It's probably obvious, but for some reason I am not getting it.  An explanation would be greatly appreciated! Or if you have a different solution, that'd be cool too. Thanks!!
Can infinitely many primes lie over a prime? If diary Tom Riddle had succeeded, would there have been two Voldemorts? I am giving a LaTeX course and have the idea to let the students find and solve problems within a few documents. Basically they shall learn how to create minimal examples and read error messages.  However, currently I have no document with 'typical' errors around.  If someone has such examples from him- or herself or within this forum I would like to have the code or link. It does not matter if its short or long. It should just be enough to let the students play with it for some minutes. Battle of Hoth: how did the AT-ATs manage to disable the shield generator? (This question is related to , and was originally posted as an answer there, but it's not the same)    In , the Empire attacks Rebel Echo Base located in Hoth. The Rebels are alerted before the attack, and raise some sort of shield to prevent orbital bombardment of their base. The Empire is then forced to deploy land forces to first disable the shield generator, before destroying the base. Presumably, the Imperial forces are deployed using dropships.  My question is about how the Rebel shield generator works, and how the Imperials manage to bypass it.  I think the shield generator is something of a plot hole. At first I incorrectly thought it was a planetary shield (i.e. a shield around all of Hoth) preventing orbital bombardment, and also preventing any dropships (with their AT-AT payload) from landing, but I think : the  states it is an energy field around just an area of Hoth, presumably Echo Base:     VEERS: My lord, the fleet has moved out of light-speed. Com-Scan has   detected an energy field protecting an area around the sixth planet of   the Hoth system. The field is strong enough to deflect any   bombardment.   So my initial objection ("how did the Imperial dropships bypass the shield?") has been answered: they didn't. They landed outside the energy field to deliver the AT-ATs. But now I have another question: does an AT-AT have to walk inside the field to fire at the generator, or can it fire into the field from outside?  Maybe the field simply cannot stop matter, only energy weapons (therefore preventing orbital bombardment), in which case the AT-ATs would be able to just walk inside. However, the script also states:     LEIA: All troop carriers will assemble at the north entrance. The   heavy transport ships will leave as soon as they're loaded. Only two   fighter escorts per ship. The energy shield can only be opened for a   short time, so you'll have to stay very close to your transports.   So it's clear that the energy field is also capable of stopping matter, in this case troop transports and fighters. It follows that it should also prevent AT-ATs from walking through it. So the only way I can see the walkers disabling the generator is by firing into the energy field!  This would mean the field is strong enough to stop matter such as ships and fighters, and energy weapons used for orbital bombardment (and also the Ion Cannon the Rebels use as an anti-orbit battery: Leia also states they must lower the shield to use it), but not able to stop whatever energy weapons the AT-ATs use!  So how on Earth did Imperial Walkers manage to disable the shield generator? I'm looking for a quick start or template, or something, to get a web site's basic layout done quickly and looking fairly professional, and then let me do the coding.  I don't want a CMS, as this is for a highly customised reporting application, but I need something like frame header and two columns and frame footer.  I can design the pretty reports for the wide right hand column myself, as the left hand column menu. How to type en-dash and em-dash in Windows 7/8? I write blog posts, and I often need to type en-dash (–) and sometimes em-dash (—) in my blog posts. Currently I keep these characters saved in a text file, and copy-paste them whenever I need them. Unnecessary to say, it is a headache to move between keyboard and mouse when you type large texts.  It would be better for me if there ware a keyboard shortcuts available for these two characters. Is there a way to type en-dash and em-dash in Windows? I use Windows 7 and Windows 8 in different computers.    Here's an aligned comparison of dashes, if you need to see the difference for some reason.   Here's-dash Here's–en-dash Here's—em-dash A few months ago I set full disk encryption during installation of Ubuntu 17.10. Now, I've decided to upgrade. The upgrade ran to the end without any problems. However, after reboot I am not able to log into my encrypted disk.  Where can be the problem? I am 100% sure I press the "right keys" on the keyboard, but technically I don't know what I am writing because of "*" characters and maybe after upgrade my keyboard layout changed. I use some characters which might be somewhere else on the keyboard. What language is default after upgrade?   By the way, I've already tried Caps-lock, but still no luck.  Help will be really appreciated. I am not a fan of installing my system and all backups over and over again. Do you need to erase the memory of *watching* a memory in a Pensieve? You remove your memory from your brain to store in a Pensieve (as Snape or Dumbledore did). Then, you re-watch the memory (e.g. as Dumbledore re-watched his memories of Voldemort's investigations with Harry) in the Pensieve.   Does that mean that you now re-created the memory in your brain (because you remember watching it) and thus you'll need to extract/erase it from your brain again if you don't wish to keep it there for security/capacity reasons (which was why you stored it in the Pensieve in the first place?  Canon answers only please. I have knowledge about 'Kile' more or less. Now  I am using 'Texmaker'. But  there is a problem in it. In Kile when we write \begin{env} it automatically write \end{env} from the following :  Settings => Configure Kile => LaTeX => Environments => making tick mark on "Automatically complete \begin{env} with \end{env}" we can set it. But in 'Texmaker' automatic environment is not working. How can I do so. Repeated squaring method How do I use the repeated squaring method to calcualte 2^176 (mod 177)? I'm not sure, but is there something about the fact that 177 is 1 greater than 176 that makes this a problem? Beamer consists of multiple frame layouts Using fontspec with Luatex it is possible to create a fake slant and use that as italic should no italics be around:  \setmainfont[ AutoFakeSlant=0.15 ]             {FontName}   However, also if an italic version exist this faked slant will be used both for \itshape and \slshape. How can I keep using the shipped italic font defined by the ItalicFont= parameter to \setmainfont but use fake slant for \slshape? Prove $$\int _0^\infty f^2(x) dx \leq \frac{2}{3}\cdot  \max_{x \in \mathbb R^+} f(x) \cdot  \int _0^\infty f(x) dx$$ for $f(x) \geq 0$ and convex.   I know via Holder's we can get without the constant $\frac{2}{3}$ but that should also be true for $f$ not convex. I tried some approaches using convexity but didn't get the desired result.   A hint or reference would also help. Also, it would be interesting to know when equality holds. Asymptotic complexity of .NET collection classes Stop all active ajax requests in jQuery What does the operator "<<" mean in C#? My Bash Prompt is currently setting the xterm titlebar using the following sequence:  PS1='\033]0;\u@\h&gt;\w\007'   Is there an easy way to display the current command in the titlebar. For example, if I am tailing a file using tail -f foo.log, I want my titlebar to say tail -f foo.log. How can I fuse a lot of cubes? So, I made a mesh with a lot of cubes ( I've created a cube, and copied it, pasted it to make figures ), And, when I've done it, I tried to smooth it, but it looks wierd because it's made with a lot of cubes, so a lot of faces, I already tried Ctrl + J, but faces remain, Is there anyway to fuse those lot of faces in one? Vim-latex adds extra brackets when typing empty brackets How to vote on friends / colleagues ' questions (or answers) Almost every time a colleague of mine asks a question, we have discussed this problem beforehand. Also, we sometimes meta-discuss the question for stackoverflow (why did I get the &quot;this might be subjective&quot; warning, is this understandable enough, etc). This is less so with answers, but I could believe for some groups this works the same. For now I'll stick to questions, but you could s/question/answer if you like. The resulting question is almost always  Interesting for me, and a problem I don't know the answer to. I probably would favorite it According to me a valid and good question (as we talked about it beforehand) I would probably want to upvote it.  Now I believe to read that &quot;ring voting&quot; is a thing, and I tried to find some questions about this, but didn't find many. I did find and read this blogpost from 2008:  . It reads to me that upvoting 'fraud' is more commonplace then downvoting and it might seem anomalous that I would upvote almost all questions from one user. If all colleagues would do this to their team, you'd get some mediocre questions getting a lot of votes, and some users getting a lot of attention, points etc from that. If you would -as seems to be done -, sort the amount of upvotes for my colleague, you'd get me being 'anomalous' in front of the crowd. On the other hand, as explained above, I do want to upvote almost all stuff some of my colleagues put on here for (I believe) the right reasons: they are questions I consider good and interesting. It's just that I know of all their questions (as they tell me about them), which obviously isn't the case for a random user. So what is the deal?  When is upvoting content from one user frowned upon? Would my usecase be in danger of being a (what I consider 'false') positive for the fraud-detection? (and would you agree with that or not) How to prevent this false positive (if you agree with me it is one, otherwise this is a moot question) -&gt; As noted in the comments: I'm not looking for tricks, so an answer could be &quot;don't vote for your colleagues&quot;. I agree that there should be no to 'tricks' around that could be used for real fraud of course