How do I show that $\mathbb{Q}[X]/(X^2 + X + 1)$ and $\mathbb{Q}[X]/(X^2 + 1)$ are not isomorphic?
I found an argument online that the two fields are not isomorphic but I can't make sense of the argument. You find it .  These are the things I'm confused about when it comes to the argument.     It says that the field $\mathbb Q[x]/(x^2+1)$ contains a root of the polynomial $x^2+1$. I suppose this is the congruence class $[x]$? But this is not the same as the zero element of the field $\mathbb Q[x]/(x^2+1)$, right?   It says that every isomorfism takes $0$ to $0$. I interpret this as the zero-element in $\mathbb Q[x]/(x^2+1)$ (which I interpret to be $[0]$) is taken to the zero-element in $\mathbb Q[x]/(x^2+2)$ denoted $[0]_2$. But in the argument it says that this means that if $\mathbb Q[x]/(x^2+1)$ has a root of the polynomial, then also $\mathbb Q[x]/(x^2+2)$ must have a root of this same polynomial. Why is that?     Greatful for any help!
Could I use a 1/2TB USB 2/3 external HDD with an XBox 360? I'm considering getting an XBox 360 (plus Kinect). I need storage for other devices as well (movies, PC games, whatever an Ouya might need), so I'm considering getting a big external HDD. Can I store downloaded games (e.g. Fez) on such a device? When he was just a professor, Dumbledore taught Transfiguration. As Headmaster what was he doing all day? In the 6th movie he was diligently looking for Horcruxes, but what about the other times?  When Fred and George pass the Maurader's Map to Harry they note that Dumbledore is pacing in his office. "He does that a lot" one of them says. Did Dumbledore just endless pace in his office as Headmaster, afraid to leave it for fear someone would know that he produced no work?   Sure, every year he had to go find a new DADA teacher, but most of those it seemed he did through correspondence. Horace Slughorn was the only one that seemed to resist. Dumbledore's 6th year may have been the only one where he earned his paycheck. I read this story in the early 2000s, but it was a Chinese translation so the English version could be older.  The beginning is a bit like the Passengers movie: on a spaceship, a man woke up from stasis too early and found that he could not go back to sleep. So he has to spend his whole life on the spaceship, alone.  For some reason (technical or ethical), instead of waking other people up, he created a fantasy adventure story featuring a brave prince as a pastime. He did stories, and drawings all over the inside of the ship. Then he died, not knowing how people are going to respond to his work. Why is sample standard deviation a biased estimator of $\sigma$? According to the Wikipedia article on  the sample SD   $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$$  is a biased estimator of the SD of the population. It states that $E(\sqrt{s^2}) \neq \sqrt{E(s^2)}$.   NB. Random variables are independent and each $x_{i} \sim N(\mu,\sigma^{2})$   My question is two-fold:   What is the proof of the biasedness? How does one compute the expectation of the sample standard deviation    My knowledge of maths/stats is only intermediate. "The train will leave" vs. "is going to leave" vs. "leaves" vs. "is leaving" Let's say I have a video running in an application.  If I hover my mouse over the taskbar, it shows a little popup windows that diplays the video  (aero peek).  Sometimes I'm running a maximized window and I want to have this little screen always popped up.  (Kind've like when your using the TV Guide menu, you still see a little clip of the video) Reference request: convergence for periodic Markov chains Consider an irreducible but possibly periodic Markov chain on a finite state space with transition matrix $P$. We know there exists a unique stationary distribution $\pi$. If the Markov chain were aperiodic, we would have $P^n_{ij} \to \pi(j)$ as $n \to \infty$. This fails if the chain is periodic, but we have convergence of the Cesaro averages: $$\frac{1}{n}\sum_{k=1}^n P^k_{ij} \to \pi(j) \text{ as } n \to \infty.$$ Can anyone point me to a reference that states this fact? Every reference I've seen only considers convergence for aperiodic chains, or "fixes" periodicity by considering a lazy version of the chain. Alternatively, is there a simple way to obtain this result using the result for aperiodic chains? I would like to use Flash Player to show web contents but since today (12 Jan 2021) Adobe is blocking it. Is there a way to use it? I've tried Firefox, Chrome, Safari and Brave browsers. Deleting comment removes "Add comment" link on Mobile If I delete a comment on the mobile version of the site (at least if it's the only comment on an answer), the "Add comment" link will vanish along with the comment, so I have to reload to leave another comment.     but got declined as , which is why I'm reposting this (as it may well be a new bug with the same symptoms).  I'm using Firefox 32.0.3 on Android. How to escape braces (curly brackets) in a format string in .NET How can brackets be escaped in using string.Format? For example: String val = &quot;1,2,3&quot; String.Format(&quot; foo {{0}}&quot;, val);  This example doesn't throw an exception, but it outputs the string foo {0}. Is there a way to escape the brackets? The following code receives seg fault on line 2:  char *str = "string"; str[0] = 'z';  // could be also written as *str = 'z' printf("%s\n", str);   While this works perfectly well:  char str[] = "string"; str[0] = 'z'; printf("%s\n", str);   Tested with MSVC and GCC. Is there a way to get DateTime value from timestamp type column? It's a well-known theorem (Corollary 8.10 in Lee Smooth) that given a smooth map of manifolds  $\phi:M\rightarrow N$ and a regular value $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a closed embedded submanifold. Is the converse true? That is, given an embedded submanifold $S\subset M$, is there necessarily a manifold $N$, smooth map $\phi:M\rightarrow N$, and regular value $p\in N$ of $\phi$ such that $S=\phi^{-1}(p)$?   Prop. 8.12 in Lee Smooth shows that this is true locally; specifically,     Let $S$ be a subset of a smooth $n$-manifold $M$. Then $S$ is an embedded $k$-submanifold of $M$ if and only if every point $p\in S$ has a neighborhood $U\subset M$ such that $U\cap S$ is a level set of a submersion $\phi:U\rightarrow\mathbb{R}^{n-k}$.   (and any level set of a submersion is of course the level set of a regular value). I feel like this is the kind of question where, if there is a counterexample, it probably is very simple, but I wasn't able to come up with one. 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? Find the value of $y(1)$ of the ODE $y'+y=|x|$. Let $y$ be the solution of $$y'+y=|x|$$ for $x\in\mathbb{R}$ and $y(-1)=0$. Then the value of $y(1)$ is    $\frac{2}{e}-\frac{2}{e^2}$ $\frac{2}{e}-2e^2$ $2-\frac{2}{e}$ $2-2e$   I don't know what to do with the absolute value function in this problem. So I started like the regular first order equation by calculating integrating factor, and got $$ye^x=\int |x|e^x dx+C.$$ Now I got stuck as how to tackle the absolute value function? Help me to solve this. Thanks! What if I accidentally run command "chmod -R" on system directories (/, /etc, ...) undelete big files - mission impossible? I've accidentaly deleted outlook.pst (6.7GB) file, while there was only 400MB free space left on primary NTFS partition (winxp).  I've tried several recovery tools to get this file back. "Ontrack Easy Recovery Pro" found 0 pst files (complete scan mode), while "Recover My Files" in sector scan mode found 5 pst's, but 4 of them of sizes from 3 to 28 KB, while the 5th one - 1Gb.  I've managed to succesfuly recover 1Gb pst file, which was 1 year old copy (the one used after the latest windows reinstall).  Now, I'm frustrated and confused   Why 1 year old file was succesfuly recovered if there were only 400MB left on primary partition? Where's 6.7GB file gone? I did some reading (), and it seems that there's almost no probability to retrieve the file I'm looking for, but wait - none of recovery tools i've used found zero-sized pst file, moreover - if due to fragmentation a file might be corrupted - we could use scanpst.exe to fix some errors and survive with 10 or 100 emails missing - whatever. Could you please recommend some more sophisticated recovery tools for this particular task?   Appretiate your help - thanks in advance I was wondering if there is such a thing as an shapefile PRJ to PostGIS SRID lookup table?  Something that can translate the most standard shapefile PRJ definitions into the likely SRID.  When using PostGIS and pgAdminIII, if you use the postgisgui to import your shapefiles, the SRID is left as "-1".  It seems like the tool should be able to parse the Esri PRJ and determine the correct (or at least a couple of options) that are the likely SRID, rather than just leave the default.  Or does the importer have the capability to reproject on the fly if you choose another SRID?  It may seem lazy on my part, but to me it seems curious that this function hasn't already been put in place.  Does anyone know if this concept is in the works, or good reason's why it has been left out? How does one change the frame size and colors in the frame option of titlesec as in MWE? Upgraded to 13.10 now clock settings are all disabled and clock does not display I've upgraded to 13.10 and now I don't have the standard menu clock, which I need for work.  I checked and 'indicator-datetime' is installed.  I even uninstalled/reinstalled that applet with no luck.  Also my "clock" settings under System Preferences are all disabled.  I can't change anything.  Does anyone know how I can get the old menu clock back?  Alternatively is there another menu clock app I can download?  Edit 1  Thanks for the responses    I've restarted several times and that didn't fix the issue. I just ran the 13.10 update today.  But I ran it again a few minutes ago.  I got   about 200KB in random updates.  The issue is still present after a reboot. # apt-cache policy indicator-datetime indicator-datetime: Installed: 13.10.0+13.10.20131016.2-0ubuntu1 Candidate: 13.10.0+13.10.20131016.2-0ubuntu1 Version table: *** 13.10.0+13.10.20131016.2-0ubuntu1 0     500 http://us.archive.ubuntu.com/ubuntu/ saucy/main amd64 Packages     100 /var/lib/dpkg/status   Edit 2  I believe the issue is related to 'rarings' changing the systray somehow...   Or , although I haven't got that solution to work yet.  Edit 3  Because of a suggestion from RolandiXor I've check the binaries.  My system is missing 'libical.so.0' but it has 'libical.so.1'.    ldd /usr/lib/x86_64-linux-gnu/indicator-datetime-service libical.so.0 =&gt; not found   Does anyone know how to get the libical.so.0 binary that the datetime service needs but doesn't install?