I have a proof that says $\mathbb Q[x]/(x^2+2)$ and $\mathbb Q[x]/(x^2-2)$ are not isomorphic.  However I feel that it is not good one...  First I see that $x^2+2$ and $x^2-2$ are irreducible in $\mathbb Q$. Then I notice that $x^2=2$ and $x^2=-2$.  Now, if we take $(a+bx)(c+dx)$ on both, we end up  $ac-2bd + (ad + bc)x$ and $ac+2bd + (ad + bc)x$.  Can I now say, that these fields do not have the same structure, so they cannot be isomorphic?  I do not want exact answer (homework :) ), but a hint would be nice to guide me into right direction.
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 :)
Unable to login after password change, having home directory encrypted showing that a group of order 45 is abelian I'm trying to understand the following proof from Dummit &amp; Foote (pg. 137) which shows why a group of order 45 is abelian. I understand everything but the last two sentences. Why is it that $|G/C_G(P)|=1$? Also, is it always true that $G/Z(G)$ is abelian? Thank you! ls -la symbolics... what does that last symbol mean? How can I subdivide this plane into even squares? Windows 10 high memory usage (unknown reason) Is multiplication not just repeated addition? If multiplication is not repeated addition, what is it? How do you define multiplication? All of us had to memorize the multiplication table in elementary school, but how did they come up with it if so many people claim that it is wrong to think of it as repeated addition? Voodoo magic? Is there a specific reason for having a very short window for passengers to get to the boarding gate before closing it? I am not a frequent traveler but it's been like this with all the flights I have been on. Here is the most recent example:  Below is my boarding pass    As you can see Gate Closes at 20:20  Here is the departures board at 18:32, clearly it says that they will announce the boarding gate at 20:10  ?!    I was hoping they will do that earlier but at 19:41 it's still the same    In fact they announced it at 20:11 but since I was freaking out and had to leg it to the gate I didn't have the time to take a photo.  You might think 9 minutes is enough, well below are the times required to reach the gates from the closest place to them     And that's for a healthy young person, no way my grandmother could be able to be at the gates within 9 minutes.  Is there a specific reason? it's not like they allocate the gate dynamically! The staff must have known about the gate hours ago as I don't expect the pilot to drive the airplane from the garage to the gate within 9 minutes? Do local flows of left-invariant vector fields satisfy $\Phi_X^t\circ L_x=L_x\circ \Phi_X^t$? Given $G$ a lie group and $X$ a left-invariant vector field. Let $\Phi_X^t$ be the local flow of $X$. Why can we conclude that $\Phi_X^t \circ L_x=L_x \circ \Phi_X^t$? Thanks! How to know number of cores of a system in Linux? I wanted to find out how many cores my system has, so I searched the same question in Google. I got some commands such as the lscpu command. When I tried this command, it gave me the following result:  $ lscpu Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit Byte Order:            Little Endian CPU(s):                4 On-line CPU(s) list:   0-3 Thread(s) per core:    1 Core(s) per socket:    4 Socket(s):             1 NUMA node(s):          1 Vendor ID:             GenuineIntel CPU family:            6 Model:                 23 Stepping:              10 CPU MHz:               1998.000 BogoMIPS:              5302.48 Virtualization:        VT-x L1d cache:             32K L1i cache:             32K L2 cache:              2048K NUMA node0 CPU(s):     0-3   In particular, this output shows:   CPU(s): 4 Core(s) per socket: 4 CPU family: 6   Which of those indicates cores of a Linux system?  Is there any other command to tell the number of cores, or am I assuming it is completely wrong? How to give a medicine with really bad taste My 4 year old daughter recently have scarlet fever. It seems antibiotics are non-negociable for it, and since she has great chance of being allergic to Penicillin, there is only one antibiotic she can get. It is not the first time we have to give it to her, and probably not the last time.  Apparently, that medicine tastes really REALLY bad. She cries a lot, panics every time we say it's time for this medicine, and vomit it most of the time, resulting in even more medicine being force into her mouth.  It is only this medicine that causes problem. She does not tantrum for other medicine, and most of them, she can take the dose we give her all by herself.  We tried reassuring her, explaining why it is mandatory to get it, explaining it will make the scarlet fever go away. We tried mixing it into other food (which seems clearly worse, taste-wise). We tried asking her to take her herself. It seems we tried everything, and the only thing that achieve it for the moment is actually forcing it into her.  We really do not want to force her in any way. Her mother cried yesterday because we had to force her, and we need to do it twice a day. It is very exhausting, and we would welcome any input or new ideas on how to give her this really bad tasting medicine to her.  EDIT : For clarification, it is a liquid medicine that is administrated with a syringe. It seems there are no alternative, although we will check with the doctor again (particularly if pills are available) How to find creation date of file? I want to find out the creation date of particular file, not modification date or access date.  I have tried with ls -ltrh and stat filename. I remember a story I have read buy could not remember the author the name or anything apart from the basic plot. The story is about some future where humanity has to fight with some aliens (not 100% sure). The story tells about a guy who was some kind of activist and who was in a prison two or more times and somehow helped humanity by being held in a prison (can't remember details). I think the story starts by saying that there is a statue in his honor that is not very well known but he did more to help humanity than famous politicians.   The story is most likely written by some famous English-speaking author, though I read it in a translation. I have studied that  and motional emf are two different lines of thinking but are essentially same. But then, how can Faraday's paradox be explained by Faraday's law of induction? Particularly in paradoxes in which Faraday's law of induction predicts zero EMF but there is a non-zero . The wikipedia article is more confusing rather than enlightening.  I need an explanation regarding  EDIT : I wanted to know how is flux changing if faraday's law is valid . Visual Studio "Find" results in "No files were found to look in. Find stopped progress." How can I draw a horizontal line spanning only some of the table cells? In a table, how can I draw a horizontal line that goes only through some of the cells? I mean a line like the one in the Foo and in the Bar area of this table:  +----+-----+ |Foo |1 |2 | |    |1 |2 | |    +--+--+ |    |1 |2 | |    |1 |2 | +----+-----+ |Bar |1 |2 | |    |1 |2 | |    +--+--+ |    |1 |2 | |    |1 |2 | +----+--+--+ Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$ Remove space around tikz How can I remove ALL space around TiKZ. So given MWE below, I want no space between fbox and the node box (so essentially I want to see 1 line). How to realize this?   Is this space in tikzpicture or in \fbox?   How to remove space between \fbox and node    MWE  \documentclass{article} \usepackage{tikz}  \begin{document} \fbox{%   \begin{tikzpicture}[node distance = 0.8cm and 0.2cm]%     \node[draw] (center) {CENTER}; %   \end{tikzpicture}% } \end{document} How to recreate subpixel anti-aliasing like Windows Command Prompt I want to be able to create a anti-aliasing effect as shown in windows command prompt with any text or vector I'd like without having to screenshot it within the program. How could I go about doing this? Let $G$ pe a $p$ group. I have to show that         the number of nonnormal subgroups is divisible by $p$       the number of subgroups differs from the number of normal subgroups by a power of $p$.   Are there any theorem that can help me to prove this ? We have discussed the Sylow theorems but I don't know how to apply them - if those are the theorems I need. (Does this theorem also hold for infinite groups $G$ ?) How do I do a search in File Explorer that includes subfolders of zip archives? In File Explorer, I can enter search terms in the upper-right search box to search the current folder and all subfolders. I can also navigate the folder tree of zip archives like any other folders. However, if I want my searches to include files in a zip archive I have to navigate to the specific folder in the zip archive where the file is located. My searches never include files in zip archives if I'm not in the archive, and when I am in the archive my searches don't include files within the archive's subfolders.  Is there a way for my searches to include zip archive subfolders?