I have to show that $\mathbb{Z}[i]/\langle 1+4i\rangle\simeq \mathbb{Z}_{17}$.  Attempt:I know that I have to use the isomorphism $\phi (a+bi)=[a]_{17}+4[b]_{17}$ but I don't know how to show that $\ker(\phi)\subseteq\langle 1+4i\rangle$.  Any ideas?
A very basic ring theory question, which I am not able to solve. How does one show that    $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$, if $a,b$ are relatively prime.   My attempt was to define a map, $\varphi:\mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ and show that the kernel is the ideal generated by $\langle{3-i\rangle}$. But I couldn't think of such a map. Anyhow, any ideas would be helpful.
In my experience, input type="text" onchange event usually occurs only after you leave (blur) the control.  Is there a way to force browser to trigger onchange every time textfield content changes? If not, what is the most elegant way to track this “manually”?  Using onkey* events is not reliable, since you can right-click the field and choose Paste, and this will change the field without any keyboard input.  Is setTimeout the only way?.. Ugly :-) Uses of the 3D Cursor I know the 3D Cursor can be used to position new objects, or snap existing ones to a certain location - but what else can it be used for?  An exhaustive list of every possible use would be great (without repeating similar things like "pivot for spin tool", "pivot for screw tool" - which could simply be said as "Pivot for modeling tools"). I have an instance of an application running in the cloud on Amazon EC2 instance, and I need to connect it from my local Ubuntu. It works fine on one of local ubuntu and also laptop. I got message "Permission denied (publickey)" when trying to access SSH to EC2 on another local Ubuntu. It's so strange to me.  I'm thinking some sort of problems with security settings on the Amazon EC2 which has limited IPs access to one instance or certificate may need to regenerate.  Does anyone know a solution? Much effort has been expended on a famous unsolved problem about the .  Not surprisingly, it's called the Riemann hypothesis, which asserts:  $$ \zeta(s) = 0 \Rightarrow \operatorname{Re} s = \frac1 2 \text{ or } \operatorname{Im} s = 0 .$$  Now there are numerical methods for approximating $\zeta(s)$, but as I understand, no one knows any exact values except at even integers, which include the trivial zeroes for which $\operatorname{Im} s = 0 $.  So I've always wondered:  For the rest, how does one prove $\operatorname{Re} s = 1/2$ exactly?  (All I know that seems vaguely useful is the argument principle, which I'm not sure helps here, but I'd be happy to learn about other techniques that aren't too far advanced.)  Edit:  Looking over this again, I found out I missed the closed-form values at .  This doesn't affect the question but seemed worth correcting.  Thanks to the people who contributed. How can I discover any USB storage devices and/or CD/DVD writers available at a given time (using C# .Net2.0).  I would like to present users with a choice of devices onto which a file can be stored for physically removal - i.e. not the hard drive. How old is Chewbacca? Best method for short time-series I have a question related to modeling short time-series. It is not a question , but how. What method would you recommend for modeling (very) short time-series (say of length $T \leq 20$)? By "best" I mean here the most robust one, that is the least prone to errors due the fact of limited numbers of observations. With short series single observations could influence the forecast, so the method should provide a cautious estimate of errors and possible variability connected to the forecast. I am generally interested in univariate time-series but it would be also interesting to know about other methods. Don't understand proof that interior of a set is open At least one prime between $\sqrt{n}$ and $n$? Prove that for all $n&gt;2$ there exists at least one prime $p$ such that $\sqrt{n}&lt;p&lt;n$ using elementary methods.  My try: If not, $\sum_{p&lt;\sqrt{n}} (\lfloor n/p \rfloor-\lfloor \sqrt{n}/p \rfloor) \geq n-\sqrt{n}$  then...? I'm looking for a site, similar to StackOverflow, but with possibility to ask opinion based questions Since opinion based questions are not allowed neither here nor on stackoverflow, I would like to ask if any of you, know some other web page (outside Stack* ecosystem), where we can ask questions like:  "Which Java Web framework would you recommend to learn"   or any other, which by nature are opinion based.  EDIT1:  Allow me to explain a bit more - I know there are webgroups, forums and mailing lists BUT most of those are for single programming language or framework and I'm looking for something more generic. Stackoverflow is great because there are experts from all sort of programming languages and frameworks, so if it would be allowed to ask:   "Best web frameworks similar to ASP.NET MVC4"  I would get a lot of opinions, from different languages and frameworks, and that exactly what I'm looking for. How to do a generalized linear model with multiple dependent variables in R? How to pass a list of columns to data.table where some are predetermined Back transforming regression results when modeling log(y) I'm fitting a regression on the $\log(y)$.  Is it valid to back transform point estimates (and confidence/prediction intervals) by exponentiation?  I don't believe so, since $E[f(X)] \ne f(E[X])$ but wanted other's opinions.  My example below shows conflicts with back transforming (.239 vs .219).  set.seed(123)  a=-5 b=2  x=runif(100,0,1) y=exp(a*x+b+rnorm(100,0,.2)) # plot(x,y)  ### NLS Fit f &lt;- function(x,a,b) {exp(a*x+b)}  fit &lt;- nls(y ~ exp(a*x+b),  start = c(a=-10, b=15))  co=coef(fit) # curve(f(x=x, a=co[1], b=co[2]), add = TRUE,col=2,lwd=1.2)  predict(fit,newdata=data.frame(x=.7)) [1] 0.2393773  ### LM Fit # plot(x,log(y)) # abline(lm(log(y)~x),col=2) fit=lm(log(y)~x) temp=predict(fit,newdata=data.frame(x=.7),interval='prediction') exp(temp)         fit       lwr       upr 1 0.2199471 0.1492762 0.3240752 When voting to migrate an off-topic question, why is the list of sites limited? Earlier, I voted to close  as off-topic.  The question has since been closed.  There is an , but it didn't show up in the list when I selected "off-topic".  The options were:   Meta Stack Overflow Server Fault Super User Webmasters Programmers   Questions:   Should Wordpress appear in this list? Are there any other sites that should appear in the list?   Should all non-beta sites appear in that list by default? What is the minimum time to consider with a connecting flight in the USA? Given the extensive time taken for immigration formalities in the US, what is the shortest time to consider between connecting flights in the US, when coming from abroad?  Until now I have always entered the US at my destination. This is not always the cheapest option. I was told that 3 hours is the advised time to go through immigration and boarding of a connecting flight. Online booking sites still offer connecting flights, with a transfer times sometimes even less then 1 hour. By the time, many files are still on my system and I don't need them anymore, so how to delete all files that are one year old at least? Silencing "Your disk is almost full" notification After upgrading to macOS Sierra, I'm getting the notification that "Your disk is almost full. Save space by optimizing storage.":    The options appear to be to store my files in iCloud, automatically delete files, or manually delete files:    My problem/irritation is that I have 80GB free of my 440GB volume.  The question: is there a way to silence this notification in a (semi-)permanent fashion, or to change the threshold it uses for the notification? Why did Mann's base explode? After the scuffle between Mann (Damon) and Cooper (McConaughey) in Interstellar, our protagonist gets a hold of a radio and alerts his shipmates to Mann's duplicity. Then his ship explodes. What happened there? Why does Loki look "sick" when he first arrives on Earth? In the Avengers movie, why is Loki sweating and weak when he first arrives on Earth, in the SHIELD base? Can't Prove formula from Sakurai's Modern QM @ Perturbation Theory I am studying Perturbation Theory from J.J. Sakurai's textbook Modern Quantum Mechanics. I am having trouble proving formulas on page 299 (5.2.5) and (5.2.6) from the previous ones [mainly (5.2.4)]. Can anyone help?  This is a problem of working with perturbation theory when the non-perturbed state has degeneracy. To elaborate I do not get why $(E-H_0-\lambda P_1VP_1)$ is not singular.  More importantly is the equation below correct and if so why:  $$(E-\lambda P_1VP_1)|l \rangle = E_D^{(0)}|l \rangle $$  That is the core of my problem. If the above equation - in the framework of Sakurai's statement of the problem - is correct I can work out all the rest on my own. However I do not get why this equation would hold true.