Let $F$ be a field of characteristic $p&gt;0$ and $a\in F$.   I have an easy question which I'm stuck on.      If the polynomial $X^p-a$ has no zeros in $F$ then is it irreducible over $F$?        Thank you!
Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]:     Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field  and let $c \in  F$. Then $x^p - c$ is irreducible in $F[x]$ if and only if $x^p - c$   has no root  in $F$. [Hint: consider two cases: char$(F) = p$ and   char$(F)$ different of $p$.]   I have attempted this a lot. Anyone has an answer?
How exactly does Knuth's Up-Arrow notation work? $$\tan(\theta_1+\theta_2+\cdots+\theta_n)=\frac{S_1-S_3+S_5+\cdots}{1-S_2+S_4+\cdots}$$  where  $S_i$ denotes the sum of product of tangent of angles taken $i$ at a time.     For example,$$\tan(\theta_1+\theta_2)=\frac{S_1}{1-S_2}=\frac{\tan\theta_1+\tan\theta_2} {1-\tan\theta_1\tan\theta_2}$$   (This formula is given  in my textbook with no derivation or background)  How to derive this? A symmetric operator has only real eigenvalues and different eigenvectors corresponding to different eigenvalues are orthogonal.   These are exactly what we want for a physical observable. I think this is the all required.   So why is quantum mechanics not satisfied with symmetric operators, but wants more? I've been wondering why I've never seen any official specifications of the maximum allowed torque that can be exerted upon lens mounts. It seems to me like a very useful specification, but I can't seem to find it from any manufacturer.  indicates that there might be no specifications for Nikon mounts. Are there really no manufacturer that publish these specification of their mounts? Just that. By good I mean something at the level of SolidWorks, AutoCAD, or similar. This is a complete limitation for me (and many others) to use Ubuntu. Is there a nice solution to get a "presenter mode" for Latex presentations? I'm curious to see some real examples of where simply capturing the same photo in RAW (and being processed by someone who can do it justice) has significantly improved the photo at the end of the process.  I understand what RAW is and why you might want to use it over JPEG, however, I'd like to actually see some examples where it has allowed for a better result. More control over tone, conversion from the more detailed data to 8-bit RGB etc.  Does anyone have or know of some processed RAW+JPEG shots for exact comparison? Basis for Skew Symmetric Matrix I'm trying to find a basis for the kernel for the following mapping:  Considering the linear transformation T: $M_{33} \rightarrow M_{33}  $ defined by $T(A) = .5(A + A^T)$. I know that this is basically asking for the basis under the condition that $T(A)=0$ which means that $A+A^T=0$ so $A^T = -A$.  I found that matrices that fit this condition are Skew Symmetric Matrices.  However, I'm not sure how to find the basis for the kernel of these matrices. WiFi problems (Atheros driver) with FreeBSD 11 With FreeBSD 10.3 my Atheros card worked fine and showed up as /dev/ath0. Now with FreeBSD 11 this is no longer the case, though the module is loaded and it seems the system recognizes the card:  root@igloo:~ # sysctl net.wlan.devices net.wlan.devices: ath0   Still, ath0 cannot be used:  root@igloo:~ # ifconfig wlan0 create wlandevice ath0 ifconfig: must specify a parent device (wlandev) when creating   What must I do to use the card? In the story I'm trying to recall, everybody on their thirteenth birthday (?) is assigned by some kind of brain analysis method a profession and given some kind of pill (?) that instantly (?) grants them the required knowledge and skills to start working immediately. Some professions are more sought after than others, especially in view of the olympic (?) games regularly held on a profession-by-profession basis.  The protagonist, however, is turned down from this system and dismissed with some general reason and has to work his way to adulthood the boring old way: studying from books. Eventually he rejects from his status as a "special" student and goes on the run, hiding from the teachers and envying the normal people.      Eventually, it turns out that the protagonist was actually smart enough that being spoonfed premade information was a waste. What he actually was assigned to was advancing the bundles of knowledge themselves, as well as being front line in the technological progress.  "Somebody has to write those textbooks! You can infuse knowledge about how to use today's technology, but you can't do that for tomorrow's - people that get assigned a profession become tied with their assigned technology and are only as relevant as that. If you want more, you must learn how things actually work, and there's no shortcut for that." (not an actual quote, but a specific passage as I remember it.)  The troubles the protagonists underwent were actually planned and necessary for him to acknowledge and accept his role in society.   This story was certainly longer than a couple pages, but it wasn't very long either. If it's a short story, it was probably written by Asimov, but I can't be sure. It rings me of Brave New World given the theme of government-issued population planning, but it certainly doesn't fit with anything else in the actual plot. Any idea? SEO preference: does a site with WWW rank better than without WWW? How can you sync Blender configs between multiple machines? I run Blender on several machines, it is a hassle to copy configs around, in order that you don't have to configure everything from scratch every time. Is there a sync functionality that does this? Merge files based on matching of first column I have two files: File1  ARS-BFGL-BAC-10975 0.9303 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11025 0.9092 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11044 0.9626 688423261 2 01/04/2015 0.9983763305 ARS-BFGL-BAC-11193 0.9544 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-10975 0.9303 688423263 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11025 0.9092 688423263 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11044 0.9626 688423263 2 01/04/2015 0.9983763305 ARS-BFGL-BAC-11193 0.9544 688423263 1 01/04/2015 0.9983763305   File2:  ARS-BFGL-BAC-10975 10 21225382 ARS-BFGL-BAC-11025 10 84516867 ARS-BFGL-BAC-11193 1 29303546   Desired output  ARS-BFGL-BAC-10975 0.9303 688423261 1 01/04/2015 0.9983763305 10 21225382 ARS-BFGL-BAC-11025 0.9092 688423261 1 01/04/2015 0.9983763305 10 84516867 ARS-BFGL-BAC-11193 0.9544 688423261 1 01/04/2015 0.9983763305 1 29303546 ARS-BFGL-BAC-10975 0.9303 688423263 1 01/04/2015 0.9983763305 10 21225382 ARS-BFGL-BAC-11025 0.9092 688423263 1 01/04/2015 0.9983763305 10 84516867 ARS-BFGL-BAC-11193 0.9544 688423263 1 01/04/2015 0.9983763305 1 29303546   So file 1 has many more rows than file 2. I only want to keep rows in the output that are in file 2 based on column1.   I have tried join but I cant get it to work right-it will tell me my files are not sorted  join -j 1 -o 1.1,1.2,1.3,1.4,1.5,1.6,2.2,2.3 &lt;(sort -k1 file1) &lt;(sort -k1 file2)   Preferably I would prefer an awk command. File 1 will be very large. I have tried   awk 'FNR==NR{a[$1]=$2 FS $3;next}{ print $0, a[$1]}' file2 file1 &gt; output   Any help would be much appreciated. Thanks  Sorry I cant comment below but just to clarify file not all rows in column1 in file 1 will be in file2.  The awk command  awk 'FNR==NR{a[$1]=$2 FS $3;next} $1 in a {print $0, a[$1]}'    will only keep the amount of rows there is in file 2. But ideally what I want is were for expample ARS-BFGL-10975 is repeated twice (realistically way more) to appear twice in my output.  Thanks for the help so far Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? Somebody asked me this question in a job interview and I replied that their joint distribution is always Gaussian. I thought that I can always write a bivariate Gaussian with their means and variance and covariances. I am wondering if there can be a case for which the joint probability of two Gaussians is not Gaussian? How does one obtain a Witcher 3 Troll Mutagen from a non-quest specific troll, at least one of which you never exchange dialogue with? I have played the Witcher 3, exchanged dialogue with 5 trolls already, and decided not to kill them. All 5 trolls were involved in specific quests. I have killed two trolls of which I did not exchange dialogue. I have yet to receive a Troll Mutagen.  How does one obtain a Witcher 3 Troll Mutagen from a non-quest specific troll, at least one of which you never exchange dialogue with? Specify a bijection from [0,1] to (0,1]. Does the lack of sleep affect my health? In the past month, I've been really busy with the university and stuff, and I don't sleep as much as I used to. I usually sleep 4-5 hours a night.  I haven't been doing that extensively ( I always kept a good sleeping routine), but can the lack of sleep affect my health on the long term ? Why is RAM not put on the CPU chip? Modern CPUs are very fast compared to all things external, including memory (RAM).   It is understandable, since CPU clock frequency has reached a point where it takes several clock ticks for an electric signal simply to run from from the CPU through the bus to RAM chips and back.   It also complicates life on many levels: multi-level cache hierarchies are built to deliver data closer to the CPU, which in turn require complex synchronization logic in the chip. Programs must be written in a cache-friendly way to avoid wait cycles while data is fetched.   Many of these problems could be avoided if a significant amount of RAM was located directly on the CPU chip. It doesn't have to an exclusive arrangement: maybe put 1-4 GB on the chip, depending on its class and allow additional memory installed separately.   I'm sure there are good reasons Intel, AMD and the like are not doing this. What are these reasons? Is it that there's no room to spare on the chip? What are the best books to study Neural Networks from a purely mathematical perspective? I am looking for a book that goes through the mathematical aspects of neural networks, from simple forward passage of multilayer perceptron in matrix form or differentiation of activation functions, to back propagation in CNN or RNN (to mention some of the topics).  Do you know any book that goes in depth into this theory? I've had a look at a couple (such as Pattern Recognition and Machine Learning by Bishop or Deep Learning by Goodfellow, Bengio and Courville) but still have not found a rigorous one (with exercises would be a plus). Do you have any suggestions? What exactly is "pressure", and what's its relation to force?