How to show that $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 5)$ are non-isomorphic as a ring?  All I could manage to  show is that,  for any isomorphism $\phi:\mathbb(\sqrt 2)\to\mathbb(\sqrt 5),$ $\phi(1)=1.$   Since $\phi$ is a homomorphism, $\phi(a+b\sqrt 2)=\phi(a)+\phi(b)\phi(\sqrt 2)=a+b.\phi(\sqrt 2)$ (Since $\phi(\dfrac{p}{q})=\dfrac{p}{q}\phi(1)$ for $p,q(\neq 0)\in\mathbb Z$) $\forall~a,b\in\mathbb Q.$
Page 105 of D. Burton's A First Course in Rings and Ideals reads     It is not difficult to show that if (I'll call them $n$ and $m$ instead of $n_1$ and $n_2$) $n$, $m$ are square free integers, then $\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ if and only if $n=m$.   Well, it is getting difficult for me, any help or would be appreciated.    My progress so far: Suppose $\phi :\mathbb{Q}(\sqrt{n})\to\mathbb{Q}(\sqrt{m})$ is an isomorphism. Then $\phi(u)=u$ for every $u \in \mathbb{Q}$, so that $\phi(\sqrt{n})^2=\phi(n)=n$. If $\phi(\sqrt{n})=a+b\sqrt{m}$ for some $a,b\in\mathbb{Q}$, then $(a+b\sqrt{m})^2=a^2+b^2m + 2ab\sqrt{m}=n$ implies $$a^2+b^2m = n$$ and $$2ab=0.$$  Then $b=0$ cannot happen since that would imply that $\sqrt{n}\in\mathbb{Q}$, where $n$ is a square-free integer. Also $a=b=0$ cannot happen. Hence $a=0$ and we have $$b^2m=n.$$ If $\psi:\mathbb{Q}(\sqrt{n})\to\mathbb{Q}(\sqrt{m})$ were an isomorphism then there would exist some $s\in\mathbb{Q}$ such that $$s^2m=n,$$ then $s=b$ or $s=-b$. Therefore the only isomorphisms from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}(\sqrt{n})$ are $\phi(u+v\sqrt{n})=u+vs\sqrt{m}$ and $\psi(u+v\sqrt{n})=u-vs\sqrt{m}$.    I don't know if there's an easier way and I don't know if I'm in the correct way here. Thanks beforehand :)
Use /testfor to make a custom command Basically, I'm trying to recreate the Faction plugins /f map in my single-player without mods as my PC doesn't run them very well. (I know it sounds strange.) I am trying to use the /testfor to test if I have written '/f map' in chat, and if I have, it would show me a custom generated map. I don't know how to word this better xD Essentially what I want is for it to test if I have written a command in chat, and if I have, display something. Why are we using a biased and misleading standard deviation formula for $\sigma$ of a normal distribution? Do you have to take the second leg of a domestic flight? How do I document a module in Python? That's it. If you want to document a function or a class, you put a string just after the definition. For instance:  def foo():     """This function does nothing."""     pass   But what about a module? How can I document what a file.py does? WordPress Installation Failed I am trying to install Wordpress on Lubuntu. I followed the instructions . Including PHP Apache and MySQL setups. Apache and MySQL appear to be running fine.  However, when I go to http://localhost/blog/, I get the following error:     Neither /etc/wordpress/config-localhost.php nor /etc/wordpress/config-localhost.php could be found.   Ensure one of them exists, is readable by the webserver and contains the right password/username. Do accepted answers still serve a useful purpose? In the Stack Exchange voting system, we have two classes of votes: standard up / down votes that anyone can cast, and special  that only the asker can.   Although accepting an answer does not always pin it at the top of the answer list, that green check mark does  about the correctness of the answer. This gives a single person much greater weight than the rest of the community. That green check mark can seem confusing to visitors, who see  or  answers marked as being correct. I've heard this from several friends recently who were surprised when they came across factually wrong answers that had been downvoted, yet still had that check mark next to them.  Moderators regularly field flags from even more experienced users who demand that the accepted answer be changed from one that is incorrect or even harmful to another. We decline these, because we're not supposed to be making technical judgments like that and because we have no way of accepting answers on behalf of the asker. However, these flags do show that some people have problems with the presentation of accepted answers vs. the community voting on all the others.  There's also all the fun that came with arguments over accept rate, and people badgering askers to accept their answers.   Questions on the Stack Exchange network are intended to be broadly useful, to help out many others in the future beyond the person asking the question. If that's our goal, why should the person asking the question have any more say than the rest of the world? If it's a broadly applicable question, why should one person be able to highlight something as correct simply because they asked it first?  I guess what I'm asking is: if we did away with accepted answers, converting them to upvotes where appropriate, and only let community voting sort out the best answers, would we lose something important? I thought I'd ask after reading  to , because I wasn't sure how I felt about accepted answers anymore. Should Community Moderators be "elected for life", or have terms? While I applaud the excellent work of , the question has come up:     How long will be the elected moderator’s “mandate”? Is it for life, or are we going to have “moderator elections” regularly?   Of course, any community moderator who wishes to recuse him or herself can certainly do that at any time.   Should we have term limits that require community moderators be cycled out periodically, so others get a chance to moderate? Should we hold regular (say, yearly) moderator elections? Is it better to have a larger pool of community moderators, and let the normal ebb and flow of turnover get us "natural" variability in moderation?   What are your thoughts on this? Knockout js loop up to a value How can I create a coffee cream effect with Photoshop? Today I would like to create a coffee cup with the cream floral on it as a text by using Photoshop. It looks like you order a cup of hot cappuccino and the bartender will draw a flower with cream on the surface of coffee but now I want to make it like a text to present a tittle.     I just need to do the cream text effect, no need to draw the whole cup of coffee since i can't use some images. So anyone can tell me how to do this cream effect on Photoshop.  Thanks in advance for any feedback! What is the typographic reasoning here? Does it make the text look less crowded? How do I uninstall xcode 5 command line tools? How far can/should one press philosophical doubt? Should we keep on questioning until nothing is left to question or is there a point on which we need to stand (which we often tend to do)? Descartes used 'I think' as this fixed point where the skepticism abates, there may be others. But what is a rational way to find one, if any? How is this question addressed in modern philosophy? How do I create a new object using Python in Blender 2.80? As part of an existing add-on, I'm trying to create an object and add it to the scene using Python. I followed a Blender Cookbook tutorial found here: . I use the following code to do so:  emptyMesh = bpy.data.meshes.new('emptyMesh') theObj = bpy.data.objects.new("object_name", emptyMesh) bpy.context.scene.objects.link(theObj)   But I get the following error when I try to link theObj to the scene objects:  AttributeError: 'bpy_prop_collection' object has no attribute 'link'   This works just fine in Blender 2.76-2.79, but doesn't work in the 2.80 test build I downloaded. Was there a change to the Python API? Or is this maybe a bug in the test build? How can I break on an unhandled exception in ASP.NET? Example of non-isomorphic structures which are elementarily equivalent I just started learning model theory on my own, and I was wondering if there are any interesting examples of two structures of a language L which are not isomorphic, but are elementarily equivalent (this means that any L-sentence satisfied by one of them is satisfied by the second).  I am using the notaion of David Marker's book "Model theory: an introduction". German language: use of comma in numbers How to get apache to send a 404 based on no matching VirtualHost section (based on ServerName) being found? I am rebuilding an old server.  Only one of the five old sites need to go live and all five of the old sites point to the same IP address.  Unfortunately any time a request is made for one of the old sites via their own unique domain names they are passed the pages from the one site that I'm trying to make live.  I would just like the old sites to display a 404 instead of the one site that I'm actually trying to make live.  As an example:   Site 1 (192.168.0.1) [www.first.com] Site 2 (192.168.0.1) [www.second.com] Site 3 (192.168.0.1) [www.third.com]   I want to move site 1 live again.  So I turned on apache on the machine with the IP address of 192.168.0.1, with all of site 1's files in the /var/www directory.  When I browse to 192.168.0.1 or www.first.com I correctly see site 1. When I browse to www.second.com and www.third.com I also receive site 1 which is the problem.  Is there a way I can 404 any request which comes in without www.first.com?  Here is my sites-available:  NameVirtualHost *:80  LogFormat "%h %l %u %t \"%r\" %&gt;s %b" common  &lt;Files ~ "^\.ht"&gt;     Order allow,deny     Deny from all     Satisfy All &lt;/Files&gt;  &lt;Directory /&gt;   Order Deny,Allow   Deny from all   Options None   AllowOverride None   Options None &lt;/Directory&gt;  &lt;Directory /var/www&gt;   Order Allow,Deny   Allow from all &lt;/Directory&gt;  &lt;VirtualHost *:80&gt;         ServerName first.com         ServerAlias www.first.com         DocumentRoot /var/www         ErrorLog /var/log/first.com.err         CustomLog /var/log/first.com.log common &lt;/VirtualHost&gt;  &lt;VirtualHost _default_:*&gt;         ServerName unknown         DocumentRoot /var/www_404 &lt;/VirtualHost&gt; This is Exercise 3.1.A. in      Suppose that $\pi: X\rightarrow Y$ is a continuous map of differentiable manifolds. Show that $\pi$ is differentiable if differentialble functions pull back to differentiable functions, i.e., if pullback by $\pi$ gives mpa $\mathcal{O}_Y\rightarrow \pi_*\mathcal{O}_X$.   Let $f: V\subseteq Y\rightarrow \mathbb{R}$ be a differentiable function on an open subset of $Y$. Let $(U,\phi)$ be a chart of $X$, $(V,\psi)$ be a chart of $Y$. Then $f\circ\pi(\phi^{-1})$ is differentiable and $f\circ\psi^{-1}$ is differentiable. We need to show that $\psi\circ\pi\circ\phi^{-1}$ is differentiable. I don't know how to prove this.   Can we say that since $(U,\psi\circ\pi)$ constructs a chart of $X$, by the compatibility, $\psi\circ\pi\circ\phi^{-1}$ is differentiable? This seems true but this does not use the fact that $f$ is differentiable.  Sorry I have no background of Differentiable Manifolds. Any help would be appreciated.  I also saw this post: , but I don't understand how $\gamma$ is smooth in the answer. $\gamma=\beta\circ\rho$, where $\rho$ is defined to be smooth, but $\beta$ is not. What is the plural of the abbreviation of "multiplicity automaton", "MA" or "MAs"? The "multiplicity automaton (MA)" is a model in compute science and its plural is "multiplicity automata". Should the plural of the abbreviation be MA or MAs?      is not a duplicate as it is focused on cases like "ATMs" where the expanded form "automated teller machines" has a regular plural ending in "s." Unlike "machines," the word "automata" has an irregular plural that does not end in "s." It's unclear from the linked answers how abbreviations that end in words like this should be pluralized. A high-ranking  mentions "VIPs," but a high-ranking  says     VIPs' is a strange one. It could be argued that the expanded acronym,   when pluralized, should be 'Very Important People' rather than 'Very   Important Persons'. 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