Just building on , where we showed that an odd number $p$ is prime in $\mathbb{Z}[i]$ if and only if $x^2+1$ does not have roots in $\mathbb{Z}/p\mathbb{Z}$.  The last part of this exercise is to show, in this case, that the multiplicative group of units of $\mathbb{Z}/p\mathbb{Z}$, denoted by $(\mathbb{Z}/p\mathbb{Z})^\times$ does not have an element of order 4, and $p \equiv 3 \mod 4$.  We can also conclude that $p \not\equiv 0,2 \mod 4$ because $p$ is odd.  Here is where I am getting a bit stuck. I assume the point is to show that if we assume that $p \equiv 1 \mod 4$, for example, $p = 5$, then $x^2+1$ has roots in $\mathbb{Z}/5\mathbb{Z}$. Here, $x^2+1 = 0 \mod 5$ for $x=2$. By the previous proposition, then $p$ is not prime in $\mathbb{Z}[i]$.  I am lacking the general statement of this proof. How can I go from assuming $p \equiv 1 \mod 4$ to $x^2+1 \equiv 0 \mod p$, in a general way?
So far, I know if $p$ is a rational prime, then $(1)$ if $p\equiv 3\mod4$, then $p$ is prime in $\mathbb{Z}[i]$. $(2)$ If $p\equiv1\mod4$  then $p=π_1 π_2$ where $π_1 $ and $π_2$ are conjugate, Then $π_1 $ and $π_2$ are primes in $\mathbb{Z}[i]$. $(3)$ $2=(1+i)(1-i)$, then $(1+i)$and$(1-i)$ are primes in $\mathbb{Z}[i]$.  What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$? For example, $-3$ are prime in $\mathbb{Z}[i]$, but not in the above $3$ cases.
When should I use # and = in ASP.NET controls? Let me know I have upvotes even after hitting the rep cap When you cap, you don't see any upvotes you get on your profile, therefore, you don't have any update on how your questions / answers are doing unless you look at them manually.   Capping the 200 daily rep is not a reason to stop posting here so could I get notified somehow that my posts are being voted on ? Maybe put a little +X in a different color letting me know that I got these votes but I won't get the reputation for it.   I can go and see in my reputation feed inside my profile and see it :    So I don't see why it wouldn't show up in the feed above.  Maybe putting the feed button green without any +X just like when you aquire a badge. Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you. Extracting extension from filename in Python Were there any improvements to the second Death Star (other than the change in exhaust port) that we know of? We know that the second Death Star was meant to have a different/non-existent exhaust port and when under construction it had a shield projected around it.  Other than these elements, one of which was planned to be totally temporary, do we know of any other improvements over its predecessor?  Did it have improved defences against one-man fighters? Did its laser charge more quickly? Did it have an upmarket cafeteria? Some areas of the mesh are dark (Shadow artefacts) I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it. What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . How do I use the chrome driver in Ubuntu 16.04? Getting image dimensions without reading the entire file How to find the best route through multiple points? I am using postgreSQL, postGIS and pgrouting. Now I am really confused on how am I going to find the shortest path between multiple points.   I know how to find it between two points through dijkstra but I am confused how to accomplish the same result with more than two points.   I have done some research and found this:    But still I did not understand if its the right way and how to install it. What exactly is "base ISO" and how do I find what is base ISO on my camera? If I understood correctly, base ISO is in full stop steps from the lowest possible ISO setting on my camera. For example if the lowest setting on my camera is ISO 100, than the following table would be base ISO:    100, 200, 400, 800, 1600, 3200, 6400   While steps in between like   125, 160, 250, 320, 500, 640, 1000, 1300, 2000, 2500, 4000, 5000   would be third-stop "pulled" values, which result more noise than perhaps higher base ISO values (for example 2500 is worse than 3200).   Did I get this right or missed by a mile, what did I miss? Do non-base ISO values generate more noise than next higher base-value ISO stop? How do I find what are my base-value ISO stops (is it the lowest ISO value multiplied by 2n)? What's the best way to use SOAP with Ruby? A client of mine has asked me to integrate a 3rd party API into their Rails app.  The only problem is that the API uses SOAP.  Ruby has basically dropped SOAP in favor of REST.  They provide a Java adapter that apparently works with the Java-Ruby bridge, but we'd like to keep it all in Ruby, if possible.  I looked into soap4r, but it seems to have a slightly bad reputation.  So what's the best way to integrate SOAP calls into a Rails app? Frequently when I ask a question I have to include quite a bit of code for reference purposes.    It would be nice to be able to bold the important lines in the code.  For example this code:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     &lt;Setter Property="IsSelected" Value="False"/&gt; &lt;/DataTrigger&gt;   Is not way long, but the important part (as seen in ) is the second to last line.  However, if I try to bold it, it ends up looking like this:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     **&lt;Setter Property="IsSelected" Value="False"/&gt;** &lt;/DataTrigger&gt;   Not what I was going for.  It would be nice to be able to draw attention to specific lines of code when needed.    (One suggestion of a different color background sounds nice too (ie a yellow background to draw the eye to the relevant code.)  If there is some other way to bold in code please let me know. How to get Microsoft Search to index common file types like .pdf and .pub I'm trying to get Windows Search (4.0) to index PDFs and .pub, .doc, etc. files.  The files are on an XP Pro PC acting as a file server. But I don't have MS Office, etc. installed (no need, trying to keep it "slim and trim") but WS 4.0 is doing the indexing of those files and not indexing .pdf and .pub.  I've found a few  (from Foxit PDF reader) rated 3/5 and only support WS 2.0 or 3.0.  Looking for other solutions and also a way to index .pub files. Why cant we replace diode connected BJTs by simple diodes in a current mirror circuit? I know that the Naruto movies are outside the timeline in the sense that they don't affect whatever happens after them. However, they must fall somewhere in the Naruto timeline — based on who are in the movies and how the characters look like, etc. For example, if Sasuke isn't in the movie, it should take place after he left. For each of the movies, approximately what timeframe (between episodes/arcs) are they set?  The movies:   Ninja Clash in the Land of Snow Legend of the Stone of Gelel Guardians of the Crescent Moon Kingdom Naruto Shippuden the Movie Bonds The Will of Fire The Lost Tower Blood Prison Road to Ninja The Last - Naruto the Movie Boruto: Naruto The Movie How to have separated but vinculated multiple Ubuntu One accounts? What should be created first in a video game? I want to run the command   python quast.py -o quast contigs.fasta   but I want to do this for multiple contigs.fasta files in multiple directories.  I'm running OSX. Whenever both my roommate and I use the wifi in a certain room in our apartment, it slows dramatically for me. It doesn't slow for him. If he uses it in another room, it's fine; if I use it in another room together with him, it's fine; it's just in this particular room. Does anyone know of a reason this could be?