If $\phi$ : $F_2[X]/(X^3+X+1)$ $\rightarrow$ $F_2[X]/(X^3+X^2+1)$ is an isomorphism, then prove that $\phi(a)=b+1$, with $a$, $b$ the classes of $X$ Consider two fields:  $K: $ $F_2[X]/(X^3+X+1)$. Let $a$ be the class of $X$ (so $a=X+(X^3+X+1))$  $L: $ $F_2[X]/(X^3+X^2+1)$. Let $b$ be the class of $X$ (so $b=X+(X^3+X^2+1))$  $F_2$ denotes the field with $2$ elements. Field $L$ is ismorphic with $K$.   If $\phi$ : $F_2[X]/(X^3+X+1)$ $\rightarrow$ $F_2[X]/(X^3+X^2+1)$ is an isomorphism, then prove that $\phi(a)=b+1$.  They give as a hint: $b+1$ is a root of $Y^3+Y+1$.  I came up with the following:  Fill in $a$ in $K$, so you get $a^3+a+1$.  Fill in $b+1$ in $L$, so you get $(b+1)^3+(b+1)^2+1 = b^3+b^2+b+1+b^2+1= b^3+b+1$, which is equal to the $a$ part above.  But I don't know if this is a way to actually prove that $\phi(a)=b+1$. How can I do this?
Constructing an explicit isomorphism between finite extensions of finite fields Suppose $K$ is a finite field, $K = \mathbb F_{p^s}$. If we take an irreducible polynomial $f$ of degree $d$ over $K$, then the splitting field $L$ of $f$ is $K(\alpha)$ where $f$ is the minimal polynomial of $\alpha$. But then $L = \mathbb F_{p^{sd}} $. Since $\mathbb F_{p^{sd}}$ is unique, we see that this is the splitting field of every irreducible polynomial of degree $d$ over $K$.   Take $K = \mathbb F_2$ and let $P(X) = X^3 + X + 1$, $Q(X) = X^3 + X^2 + 1$. Let $L$ be the splitting field of $P$ and $L&#39;$ be the splitting field of $Q$. The above tells us that $L$ and $L&#39;$ are isomorphic. I would like to construct an explicit isomorphism between $L$ and $L&#39;$.   I know that $L \cong \mathbb F_2[X] /(X^3 +X + 1)$ and $L&#39; \cong \mathbb F_2[X] / (X^3 + X^2 + 1)$. Intuitively, I want to find an isomorphism $\phi : \mathbb F_2[X] \to \mathbb F_2[X]$ such that $\phi((X^3 + X + 1)) = (X^3 + X^2 + 1)$. A little playing around gives me $\phi(X) = X+1$. It now feels like I'm falling at the last hurdle: how do I finish the construction of an isomorphism between $L$ and $L&#39;$? I don't think $\phi$ makes sense as a map from $L$ to $L&#39;$, yet it seems the map I want.
Often called "He-Who-Must-Not-Be-Named" and "You know who", was it common knowledge at the Ministry of Magic that Voldemort was the same person as Tom Riddle? Is the below brief explanation correct and complete (at least in context)? Can it be improved/simplified without getting technical? I am especially concerned about the last sentence: Is that reasoning absolutely clear to a layman?            The problem can be viewed as drawing the above graph without lifting your hand and without retracing a line. Observe that each dot is connected by three lines. Also observe that you have to draw a line to arrive at a dot, and you have to draw a line to leave that dot. Because each dot is connected by three lines, each must be visited twice. But it is impossible to visit each dot twice (even number of times) and end up with each dot connected with three lines (odd number of times). I have a solid understanding of most OOP theory but the one thing that confuses me a lot is virtual destructors. I thought that the destructor always gets called no matter what and for every object in the chain. When are you meant to make them virtual and why? The question is about the encryption equation in asymmetric encryption.  For $c\equiv m^e\bmod n$, prove that for every unique $m$ there is a unique $c$.  Here $n=pq$ where $p$ and $q$ are primes larger than $m$ and the exponent $e$, is $\geq3$ and coprime with $p-1$ and $q-1$.  Thanks! Placing however at the start of a sentence vs placing it in the middle of a sentence? Basically, what's the difference between this:     However, it didn't mean that I didn't have the potential to become a   rock star.   and this?     It didn't mean, however, that I didn't have the potential to become a   rock star.   Or does it mean exactly the same thing? I thought this would be really simple, but it's presenting some difficulties. If I have std::string name = &quot;John&quot;; int age = 21;  How do I combine them to get a single string &quot;John21&quot;? I am working on a large C++ project in Visual Studio 2008, and there are a lot of files with unnecessary #include directives. Sometimes the #includes are just artifacts and everything will compile fine with them removed, and in other cases classes could be forward declared and the #include could be moved to the .cpp file.  Are there any good tools for detecting both of these cases? How do I copyright my website? Hyperref warning - Token not allowed in a PDF string Where can I find documentation on formatting a date in JavaScript? Quotient rings of Gaussian integers How to install Intel CedarView drivers on Ubuntu 12.10 or higher? I am looking for instructions on how to install cedarview graphics drivers on Ubuntu 13.04 32bit.  My graphics hardware is a GMA3600 aka Power VR SGX545.  EDIT: It seems that they are not in Raring yet, but only Precise:   EDIT2: they seem to be closed source. With the advent of the WordPress Answers Stack Exchange site, there's undoubtedly going to be overlap.  But I just want to get some advice:  I'm a developer and I kinda hate WordPress, but have to use it occasionally. My WordPress questions are usually development ones (when I have them), and most often related to 'how WordPress handles things', which are pretty obvious in straight programming terms.  But with the WordPress layer of abstraction, they become problematic.  I find that most developers that have gone near WordPress are usually also PHP/JS/CSS/MYSQL guys who dabble with WordPress and, as a result, hangout more on SO. And theres a die hard bunch who are heavy into the WP.SE site, but generally the WP site is for people trying to muddle through the day to day operation of using WordPress and its many themes/plugin etc.    As a result, if I were to ask something pretty in-depth about writing / integrating new &amp; strange features or something, I'm talking to a much smaller group of devs than I would be on SO.  So, enough of the ramble, my question:  Should I be posting heavy development questions relating to WordPress on the WP.SE site or to SO? Can I post to both, or is cross posting frowned upon in these cases still? Zooming in to inspect texture details in 3D view clips some closer faces like shown in the image. It's not the camera view so changing camera settings doesn't help.   Where can the distance to the clipping plane be adjusted? Recently I had a bunch of my own scripts for the thumbnails in Nautilus: , , office documents and so on.  Seems like either I’m doing something wrong (thumbnail generation behavior changed), or custom thumbnailes don’t work in latest GNOME at all, even my previous scripts, which worked smoothly on previous versions of Ubuntu.  I’ve done some research, tried to run a simple script (see below) and it didn’t work out.  My test thumbnailer in /usr/share/thumbnailers/z.thumbnailer:  [Thumbnailer Entry] Exec=/home/maximal/thumb.sh %s %i %u %o MimeType=image/webp;image/x-webp   Where /home/maximal/thumb.sh is:  #!/bin/bash echo $0 &gt;&gt; /home/maximal/thumb.log echo $1 &gt;&gt; /home/maximal/thumb.log echo $2 &gt;&gt; /home/maximal/thumb.log echo $3 &gt;&gt; /home/maximal/thumb.log echo $4 &gt;&gt; /home/maximal/thumb.log   Then, when I remove thumbnail cache rm -rf ~/.cache/thumbnails and restart Nautilus nautilus -q, my log file doesn’t contain anything and ~/.cache/thumbnails/fail/gnome-thumbnail-factory contains all the failed thumbs (looks like corrupted/empty PNGs):    So, apparently, Nautilus even didn’t try to execute my thumbnailer script. When I try to run thumb command itself, it works perfectly. For example convert through cwebp|dwebp package:  convert -thumbnail 256x256 file.webp png:file.webp.png   When I wrap this command to a thumbnailer, it looks like it wasn’t even requested to run by Nautilus.  [Thumbnailer Entry] Exec=convert -thumbnail %sx%s %i png:%o MimeType=image/webp;image/x-webp;   Exactly the same situation occurs with other my thumbnailers: audio waveforms, office documents, photoshop files and so on.  Could anyone help? How do I change permissions on a FAT32 formatted drive? How many combinations of groups are there where no member of a group has been with another member before? I found this hard to word in the title, so let me give an example. I have 16 students, and I want to split them up into 4 groups of 4. However, I want to make sure that every time I have a new combination of groups, that a student has never been with any of their new groupmates before. I found info on how to separate objects into groups, but not in only unique ways as I've described above. I think I figured this out the rote way, by assigning each student a number and then finding the unique combinations, and got 3. But I'd like to know the math behind it, and how it could be applied to other configurations (20 students w/ 4 groups of 5?). Thanks for any help you can give! How to terminate a script? I am aware of the die() command in PHP which exits a script early.  How can I do this in Python? What can I do to prevent passport stamps being put on blank pages? I'm running low on blank passport pages which are sometimes needed for full page visas such as those issued by Armenia, India, and Vietnam.  When I crossed the border into Turkey the other day I got my visa-on-arrival and stamp, but instead of putting it on a partly used page, which I have many, the guy at customs and immigration put them on a previously blank page. Now I have one fewer!  What can I do to prevent or at least reduce the chances of this happening?  It's worth remembering that there's likely to be a language barrier and it's always best to be ultra-nice to these border officials.  Has anybody heard of any trick such as sticking bit of paper in your passport, either blank, or looking a bit like visas? I have noticed that nobody ever stamps a page that has a remnant of something once stapled in there in Japan. There is a kind of stub that had its major part torn off along a perforation. Might I be able to emulate this somehow or would that even be worse making the blank pages invalid for full page visas due to having holes from staples? I know some things can get very picky when it comes to customs and immigration. Find the MLE of the unknown parameter $\theta$ when $X_1,X_2,...,X_n$ is a sample from the distribution whose density function is: $$f_X(x) = \frac12e^{-|x-\theta|}, -\infty&lt;x&lt;\infty$$ What I did so far: $$\text{maximize }(\frac12)^ne^{-|x_1-\theta|-|x_2-\theta|-...-|x_n-\theta|} \\ \text{thus we want to minimize }|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|$$ Since $|x_i-\theta|$ is positive for all $i$, we want to solve for $\theta$ so: $$|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|=0$$ What do I do from here?