$x^p - t$ has no roots in $\mathbb{F}_p(t)[x]$ I am actually trying to prove that $x^p - t$ is irreducible in $\mathbb{F}_p(t)[x]$. My approach is to consider a splitting field $E$ so that: $$x^p - t = (x-\alpha_1)\dots(x-\alpha_p)=(x-\alpha)^p   \ \ \ \ \text{for some $\alpha$ in $E$}$$ Then assuming by contradiction that $x^p - t = f(x)g(x)$, then $f(x) =(x-\alpha)^r$ with $r&lt;p$. This implies that $\alpha \in \mathbb{F}_p(t)$. Should this be a contradiction right? Why can't $\alpha$ be here?
Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$ I don't think I fully understand.  Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number.  Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. We can assume $\gcd(P,Q)=1$  and $x_0^p-t= \frac{(P(t))^p}{(Q(t))^p}-t= \frac{P(t)^p-tQ(t)^p}{Q(t)^p} =0$, so coefficients of $P(t)^p, tQ(t)^p$ must be identical, which contradicts $\gcd(P,Q)=1$,   hence such $x_0$ does not exist and the polynomial has no root in $K$.   am I about right?    anyway, I would appreciate an explanation about $\mathbb{F}_p(t)$, what is $t$? what is the meaning of a variable which does not belong to any specific "world"? I cannot use $t$ as if it was a member of $\mathbb{F}_p$ and hence cannot assume $t^{p-1} = 1 \pmod p$...
Got NODATA issue: 'NODATA' (does the network require authentication?) How do you split a list into evenly sized chunks? The is sometimes pronounced &quot;tha&quot; (/ðə/) or &quot;thi&quot; (/ðiː/). Which is the correct pronunciation of this word? Are both correct and used interchangeably at specific places? If the second question is correct, please provide the rule of pronouncing at different places. The definitive guide to ConTeXt MkIV documentation I'm currently trying to extract textures from a binary file, and ideally I'd like to save each texture to a file or embed in the .blend file. However, before saving an image, I need to be able to create an image in memory from an array of pixel data.  The PIL library contains the kinds of functions I need to use, but it is not included with Blender. Is there anything similar to PIL in Blender? bpy.ops.image seems to be close to what I need, but lacks necessary pixel set/manipulation functions. I was wondering what is the purpose of piping the downloaded file by curl into sudo -E bash - in the following shell:   curl -sL https://deb.nodesource.com/setup_5.x | sudo -E bash - Number formation from given digits I was asked this question in a recent interview. You are given two digits $0$ and $9$. Using only these two digits you must form the least positive number that is a multiple of a given '$n$' . The value of $n$ will be provided before hand. For example if we are given $n$ as $7$ then the answer will be $9009$ .  I tried using some basic divisibility principles but was not able to come up with a correct solution and code it up. Probably I need some help in number theory Creating or referencing variables dynamically in Sass I want to know where the Metro applications (Mail, People, Messages, ...) executable file path is.  Also, how can I run Mail application when my PC starts? node.js conflicts: /sbin/node vs /usr/bin/node I have two Ubuntu 12.10 machines: machine A is a VMWare VM and machine B is and old Acer laptop. On both machines, I installed node using apt-get. However machine A installs the main binary as /usr/bin/node and machine B as /usr/bin/nodejs. Here are some behaviors I observed:   Both machine returns /usr/bin/node in response to the which node command. However, there is no such file on machine B. Issuing the command node works for machine A, not B. On B, I have to use nodejs instead. On B, there is a binary called /sbin/node, A does not have it.   These behaviors causes inconsistency problem when developing on both machines. For now, I worked around by renaming the binaries on B as such:  sudo mv /sbin/node /sbin/node-sbin sudo ln -s /usr/bin/nodejs /usr/bin/node   Is there a way to really fix this, so that node will be installed as /usr/bin/node instead of /usr/bin/nodejs? How to remove accents / diacritic marks from a string in Qt? How to determine $\int x^2 e^{-x^2}\mathrm dx$? Fastest way to move Collection from Sharded MongoDB Cluster to a another Sharded MongoDB Cluster Benchmarking hosting providers IO with Bonnie I read a book of this series probably around 2000. I think the series was quite new then but not sure.  The main thing I remember about it is that magic is based on words of power. If you knew one word you got some extra ability, knowing two words gave you yet some other ability, and if you knew three words you became a very powerful wizard. Only one person at a time could know each word, and telling someone your word meant you died.  The main protagonist somehow gets hold of one word and then for some reason strives to learn more.  EDIT: I think that I accidentally picked up the second book of the series and never read the first one. From what I remember, the protagonist learned one word of power and got some kind of supernatural ability (cannot remember what though). He then ends up in some kind of challenge where he has to cooperate with another person who knows another word that gives super-strength.  Not very coherent memories, but I hope someone knows what series it is. It has been bugging me for years. (Not even sure if the series is good at all, but still would like to know the name.)  EDIT2:  I remember the world as being medieval-ish. I do not think there were dragons, but there may have been other humanoid races (I have a very weak recollection that there may have been an orc or similar in the story, but this could be completely wrong). The protagonist was male. I have a box that I have extruded to the shape I want. Then I placed a cylinder so that it intersects the box. After that I deleted the bottom faces of both the cylinder and box.  Now what I want to do is delete the portion of the cylinder that is inside the box and join the nodes so that it is one object. I have already joined the meshes so I can select points/edges/faces from either shape. It seems like the easiest way would be to just put nodes where the edges of the cylinder and box intersect and then delete the parts I don't care about. I can't find a good way to do that though.  What's the best way to accomplish what I'm trying to do? Do the search engines take the number of 404 errors returned by a site into account when calculating the site rank? For example I have 8,000 404 errors and I am thinking that that's affecting my SEO in a negative way.  However, I read  about 404 errors which suggests that they are not always a negative quality indicator.   So I am confused...are high numbers of 404 errors harmful to a site or not? Enable Wireless Monitor mode for traffic sniffing I need to sniff all the wifi traffic that I can capture using wireshark in Ubuntu. My wireless interface is named as eth1 in the interfaces list. When I select that interface in Wireshark and start new sniff, I got following error;    As  thread says, I tried to installed the mentioned driver but I got an error in installation saying that that driver is not supported by my device. After that I tried the second answer in the same thread and run following command to enable monitor mode in my wireless card.  echo 1 &gt; /proc/brcm_monitor0   Then I saw a new Ethernet interface (not a wireless interface ) called prism0 in wireshark interface list. I couldn't start a sniff using that interface using monitor mode because in that interface settings, monitor mode check box has been disabled.  I have installed a broadcom wireless driver in Additional Drivers as follows:    Is there an error listing my Wifi interface as eth1 rather than wlan0 in the interfaces list? How can I start my wireless interface in monitor mode for sniffing?   Thank you. How do I install the latest version of EMACS and pick the appropriate UI? I'm running the latest Kubuntu (12.04 beta 2) and I would like to run the latest emacs (currently v24). The repos are one version behind. What's the best way to install v24 or later (and avoid future version conflicts)?  Also, is there any reason not to aways use the GUI version of emacs if X is running? For example, could I set the GUI emacs version as the default text editor and use it to edit cron jobs (crontab -e)? I'm assuming the answer is yes, but since I haven't done that yet (my default editor is nano), I want to check if there are reasons I should leave nano as the default editor.  Usually when I'm working on the command line I end up using nano. Now that I think about it, I have no idea why I keep doing that. Is there any downside to calling a GUI editor when working in an X terminal?  EDIT: I briefly tested these two versions   GNU Emacs 24.0.94.1 (x86_64-pc-linux-gnu, GTK+ Version 3.3.20) from  GNU Emacs 23.3.1 (x86_64-pc-linux-gnu) installed by default in Kubuntu.   .  Unfortunately (for me) the defaults installed version (23.3.1, 23.3+1-1ubuntu9) is the nox version.   Package: emacs23-nox Status: install ok installed Version: 23.3+1-1ubuntu9 Replaces: emacs23, emacs23-gtk, emacs23-lucid   The package with version 24 opens in GUI mode by default. That's what I prefer. Some of the version 24 changes that interest me are listed in the references below.   But there appear to be a multitude of different packages and versions I could install.  References:   What’s New In Emacs 24 (part 1) | Mastering Emacs  "    shell-mode uses pcomplete rules, with the standard completion UI.  Yowzah! There’s a lot of cool, new functionality hidden away in this gem of a change." EmacsWiki: Recent Changes Apple has introduced , also known as "rootless", with OS X 10.11, El Capitan. I understand this is a step for general protection against malware but as a developer I need write access to some of the files it locks away.  How do I disable this protection?