Compute $\mathrm{Aut}(S)$ of the ring $S=\mathbb{Q}[x]/(x^2)$ 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.
Determine $\text{Aut}(S)$ where $S = \mathbb{Q}[x]/(x^2)$ 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!!
Drift velocity in Drude model Find all integral solutions of $ x^4 + y^4 + z^4 -w^4 = 1995 $ Why is this quotient space not Hausdorff? When does $120$ divide $n^5 - n$ Find n such that $n^5 - n$ is divisible by $120$. (*Here $n$ belongs to the natural number set)  Seems a trivial problem, but the answer mentioned in the text book seemed incomplete to me so I'm here just to ensure whether approach to this problem is correct or not.  So my approach is as follows: We know that $n^5 - n$ can be written as $n^5-n = n(n-1)(n+1)(n^2 + 1)$  Now since it's product of three consecutive natural numbers so it's divisible by $3! = 6$. Now using Fermat's Little theorem we can prove that $n^5= n \mod5$ hence $n^5 - n$ is divisible by $5$ as well. So $n^5 - n$ is divisible by $30$. Now to ensure their is another factor of $4$ in the term we need to ensure that $n^5 - n$ is divisible by $8$. I found that all odd natural numbers and multiples of $8$ makes $n^5 - n$ divisible by $8$, hence $n^5 - n$ is divisible by $120$. But in the text book they have only mentioned odd natural numbers as answer, which seemed incomplete to me.  Thanking everyone in advance:P For a given hardware configuration, how do I find out if Ubuntu will run on it? What considerations should I take into account when choosing an Ubuntu version and  such as:    with a lighter desktop than the usual Gnome and Unity  with the even lighter LXDE desktop   Obviously Ubuntu does not run on some processor architectures. So how do I go about choosing the right version and derivate. How can I find out the minmal system requirements? Configure Postfix to use external MX servers for delivery of local mail if user is unknown If object A is moving at velocity $v$ (normalized so that $c=1$) relative to a ground observer emits object B at velocity $w$ relative to A, the velocity of B relative to the ground observer is  $$ v \oplus w = \frac{v+w}{1+vw} $$  As expected, $v \oplus 1 = 1$,  as "nothing can go faster than light".  Similarly, $v \oplus -1 = -1$. (same thing in the other direction)  But what if object A is moving at the speed of light and emits object B at the speed of light in the exact opposite direction? In other words, what is the value of $$1 \oplus -1?$$ Putting the values in the formula yields the indeterminate form $\frac{0}{0}$. This invites making sense of things by taking a limit, but $$ \lim_{(v,w)\to (1,-1)} \frac{v+w}{1+vw}$$ is not well-defined, because the limit depends on the path taken.  So what would the ground observer see? Is this even a meaningful question?  Edit: I understand $1 \oplus -1$ doesn't make sense mathematically (thought I made it clear above!), I'm asking what would happen physically. I'm getting the sense that my fears were correct, it's physically a nonsensical situation. Java Command line arguments I am trying to detect whether the 'a' was entered as the first string argument. I want to install Windows XP on my desktop computer (only XP), but my CD drive is not working.  I've only found installation tutorials on how to install it on the pen drive rather than from the pen drive.  How can I install Windows XP from a USB flash drive? Careers Stack Overflow - Tag "ruby-on-rails" returns jobs that don't contain the tag Laser positioning on barn door tracker Editing the "Self-Learner" Badge Description Stemming from  the text for "Self-Learner" badge is pretty easy to get confused by.  Current Text     Answered your own question with score of 3 or more    Issue  The person in the above question was confused as they thought they needed to answer their own question, and that the question had to have a score of 3 or more. What it actually means is that they have to answer their own question, and the answer needs to have a score of three or more. The way it is phrased is ambiguous, so why not make it clear?  Suggestion  Edit the text as follows:     Provide an answer with a score of 3 or more to your own question   No more confusion. How can I draw simple trees in LaTeX? I need to draw a tree, and constantly using dotty and including the PNGs is starting to be a pain. Is there a way to get LaTeX to draw it's own trees that doesn't involve learning an entirely new language like TikZ? What's the difference between a shunt resistor and a four-terminal resistor? I want to know if the four-terminal resistor and the shunt resistor are the same.   Is there another use for the four-terminal resistor except measuring current? I would like my JSON output in Ruby on Rails to be "pretty" or nicely formatted.   Right now, I call to_json and my JSON is all on one line.  At times this can be difficult to see if there is a problem in the JSON output stream.  Is there way to configure to make my JSON "pretty" or nicely formatted in Rails? I have made a command  \newcommand{\mysec}[2]{%  \textbf{#1} \\ #2 }   I use it with \mysec{A}{B}.  But sometimes I also want a third parameter. I could make a whole new command for those situations where I need 3 parameters. But I thought if I could handle it with an optional parameter in my original command?  So I have to check if the third parameter is set and how to print if it is set. Should my flair be including sites with less than 200 reputation? I took a look at my profile, where I stuck my flair. Here's the flair:    However, I don't have 28k rep. According to chat, I have 19.7k.  Note: I know about the combined flair - I've had this in my profile for at least a year. It had been showing me the same as in chat, now it isn't.   What is going on here?  Edit: I'm reporting a bug here, that the combined flair is not showing the correct number, as it has in the past. Capacitor on USB cable? I was messing around with a USB camera. I wanted to change the cable because is trimmed but as I was cutting it I saw a small thing (image below). As I was trying to cut the casing and see what it has inside I saw a spark. I tried to connect it with another cable but I don't know if I'm gonna damage the camera or the USB port.    What is it and can I connect the camera without it? Is it possible to turn off the Two-Factor Authentication for an Apple ID? Is it possible to turn off the Two-Factor Authentication for an Apple ID?  I have more than one device linked to this Apple ID and I know the password and security questions. "The train will leave" vs. "is going to leave" vs. "leaves" vs. "is leaving" From the grammatical point of view all are correct, just the meaning are different, please bring your clarification, thank you.        The Train will leave at 10:00 tomorrow morning.   The Train is going to leave at 10:00 tomorrow morning.   The Train leaves at 10:00 tomorrow morning.   The Train is leaving at 10:00 tomorrow morning.