Is there a known proof without differentiating that proves that all irreducible polynomials over $\mathbb{Q}$ are separable? (Or even better, for all fields of characteristic $0$.)  EDIT: As people seem to question this thread; I do know a proof with derivatives - my motivation for one without is simply curiosity. Multiple approaches are always nice.
I am reading separable extension. I don't quite understand why $\text{char} 0$ fields must be perfect. If $F$ is a $\text{char} 0$ field, $f(x)$ is an irreducible polynomial in $F[x]$. Then why $f$ must be separable? Assume $f$ has multiple roots, say $r$ is a multiple root of $f(x)$. Then $r$ is also a root of $f'(x)$. $r$ is in the splitting field of $f(x)$ over $F$, denote it as $E$. Then why $f$ must divide $f'$? I know $f$ is irreducible but I don't know how the argument goes on. Thanks for any help!
How to calculate $\lim \limits_{n \to \infty}(1 +\frac{a}{n})^n$? I know what solution is $e^a$ but I don't know how to calculate this limit:  $$\lim \limits_{n \to \infty}\left(1 + \frac{a}{n}\right)^n=e^a$$  Can someone explain the steps to me? Replacements for switch statement in Python? I want to write a function in Python that returns different fixed values based on the value of an input index.    In other languages I would use a switch or case statement, but Python does not appear to have a switch statement.  What are the recommended Python solutions in this scenario? Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares Using induction, how can I show the following identity about the fibonacci numbers? I'm having trouble with simplification when doing the induction step.  Identity: $$f_n^2 + f_{n+1}^2 = f_{2n+1}$$  I get to:  $$f_{n+1}^2 + f_{n+2}^2$$  Should I replace $f_{n+2}$ using the recursion? When I do that, I end up with the product of terms, and that just doesn't seem right. Any guidance on how to get manipulate during the induction step?  Thanks! How do I make a bootable USB OS X installer on a Mac? Does anyone have a Terminal command-line recipe for making a new bootable USB flash drive for one of the new MacBook Air's?  Is there a way to dd copy the USB boot device that comes in the box and make the new copy bootable?  I'd like to make a new backup copy of the USB rescue device for my MBA, preferably on a much larger flash drive with room for a backup of some of my working directories.  Added:  And is it possible to make, not just a rescue drive, but a bootable USB drive that will boot a fully updated version of the OS (say 10.6.8 or later)?  If so how on a MacBook Air 11, and how big a USB stick will this require? Tight bound for $T(n) = T(n^{1/2}) + 1$ Considering that sound can't travel through vacuum. Navigation bar is too long on progressbar theme, can it wrap? My navigation bar at the top of each slide (that links to each section) has too many sections. I would like to keep the bar and keep each section but I would like to wrap it i.e. have it on two lines.  Is that possible?  I am using compress and I am using  as a theme. Do known-plaintext attacks exist for public key encryption? Possible to do a MySQL foreign key to one of two possible tables? Luggage transfer in Dubai between flights, without a visa I am traveling from Stockholm to Dubai and Dubai to Nairobi using two different airlines. I have Kenyan nationality and can't go through Immigration because I don't have a Dubai visa.    How I will get my luggage? Do fontsize environments exist? Without any doubt there are fontsize switches in the form of control sequences (described  for instance.) But, from time to time I ran into code examples where environment versions of those were used, e.g. \begin{small}...\end{small}. After testing those successfully I began to incorporate them into my TeX vocabulary.  A few days ago @GonzaloMedina commented on one of my answers that those environments do not exists and constructions relying on them work by pure coincidence. Then I tried to find out more about those ominous environments. Ominous because they don't seem to be somehow documentated. (The only thing I found that tried to explain them in a somehow systematical manner was some Wikibooks resource.)  Are these environments real or not? Order of redirections I don't quite understand how the computer reads this command.  cat file1 file2 1&gt; file.txt 2&gt;&amp;1  If I understand, 2&gt;&amp;1 simply redirect Standard Error to Standard Output.  By that logic, the command reads to me as follows:    concatenate files file1 and file2.  send stdout from this operation to file.txt. send stderr to stdout. end?   I'm not sure what the computer's doing. By my logic, the command should be   cat file1 file2 2&gt;&amp;1 &gt; file.txt   but this is not correct. how to achieve the journal style shown in the screenshot? Means help me i want to write these all things with perfect setting.. I'm looking for a site, similar to StackOverflow, but with possibility to ask opinion based questions Since opinion based questions are not allowed neither here nor on stackoverflow, I would like to ask if any of you, know some other web page (outside Stack* ecosystem), where we can ask questions like:  "Which Java Web framework would you recommend to learn"   or any other, which by nature are opinion based.  EDIT1:  Allow me to explain a bit more - I know there are webgroups, forums and mailing lists BUT most of those are for single programming language or framework and I'm looking for something more generic. Stackoverflow is great because there are experts from all sort of programming languages and frameworks, so if it would be allowed to ask:   "Best web frameworks similar to ASP.NET MVC4"  I would get a lot of opinions, from different languages and frameworks, and that exactly what I'm looking for. has two corollaries:   The gravitational attraction of a spherically symmetric body acts as if all its mass were concentrated at the center, and The gravitational acceleration inside the cavity of a spherically symmetric body vanishes.   Consider a spaceship floating freely in space. In a homogeneous universe, the combined attraction from all matter should cancel out, and the spaceship should stay motionless. Nevertheless, I'm free to divide the attraction into several parts originating from different parts of the universe: In the figure below, I've divided the universe into a red sphere centered on some arbitrary point (×) with my spaceship located at the edge of the sphere, plus infinitely many shells centered on the same point.  By corollary #1, the gravitational attraction of the red sphere equals that of all its mass centered at the point ×. By corollary #2, the combined acceleration of the spaceship from all mass in the green shell vanishes. The same can be said for the blue shell, the orange shell, and so on ad infinitum.  Hence my spaceship should start accelerating toward ×. By choosing the sphere large enough, I should be able to make it accelerate arbitrarily fast, and by choosing the location of × I can make it accelerate in any direction.    Of course this doesn't work, but why?  My best guess is that, even in an infinite universe, you can't keep adding spheres because you'll exit the observable universe, in which case there's no way to feel the gravity in part of the shell so that it's no longer symmetric. Perhaps also the expansion of the universe matters. But see the last two points below.  A few more things to consider:   The mass of the red sphere increases with the chosen radius $r$ as $r^3$, while the acceleration it generates is proportional to $r^{-2}$; hence the acceleration increases linearly with the chosen $r$. Our universe — the "Universe" — has an average density of some $10^{-29}\,\mathrm{g}\,\mathrm{cm}^{-3}$. Hence if I set $r$ equal to the radius of the observable Universe (46.3 billion light-years), the acceleration is a minuscule $10^{-7}\,\mathrm{cm}\,\mathrm{s}^{-2}$. If that bothers you, choose another universe where $\rho$ is ten orders of magnitude higher. Our Universe is not really homogeneous, but on large enough scales ($\gtrsim$ half a billion light-years) it seems it is. Still, the acceleration of the spaceship will be dominated by nearby sources. If that bothers you, choose a sufficiently homogeneous universe. On the scales we're considering, the Universe is not governed by Newtonian dynamics, but by general relativity. If that bothers you, use  instead — I think the issue is the same. If the issue is really that the size of the observable universe matters, then my intuition tells me that I can just choose an arbitrarily old universe where the asymmetric contribution from the most distant shells is arbitrarily small. If the issue is that the universe expands (so that gravity from the far side of a shell is somehow weakened, or "redshifted"), then my intuition tells me that I can just choose a sufficiently static universe. This is irritating me. I seen several suggestions (all using different files and syntax) and none of them worked.  How do I set an environment variable for a specific user? I am on debian squeeze. What is the exact syntax I should put in the file to make ABC = "123"? How to install Flash player on Ubuntu? Even though I have the latest Adobe Flash player 11.2 for Linux in Firefox -   - a particular online flash game still requests that I update to the latest flash player.  I've tried this online game on my laptop (Windows 8 with Flash Player 11.8) and it has no problems.  Is there a way to trick Ubuntu or this game into thinking Flash Player 11.8 is "installed"? Is there a way to force Google Chrome to remember passwords for sites that have autocomplete turned off? Clarification: I am a programmer, but ideally, I'd like a method that I could help regular users implement, too. Nonphysical entities cannot be observed. Therefore such entities cannot be verified by observation. How could statements like "God exists" be even considered true? Why would anyone appeal to the metaphysical realm at all?  It seems to me that the best method by which we should go about in understanding the world is by initially ruling out the possibility of the metaphysical altogether for the simplest answer (a sort of null hypothesis). Once we can get a hold of a natural cause of some phenomenon, we should be satisfied that that is all there is to it. Formulating some nonphysical cause would be unnecessary, superfluously complicated, or even cluttered. Now if there was some problem that we were certain could not be solved by observation or experimentation or whose solution could not possibly exist in the observable realm, yet must be true, I would consider the possibility of a metaphysical cause.      That said, I don't think that if you can't empirically observe something, it doesn't exist; rather, I would declare total agnosticism and say that it is unknowable. Number of permutations of $n$ elements where no number $i$ is in position $i$ I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is not because 5 is in position 5.  I know that the number of total permutations is $n!$. Is there a formula for how many are acceptable given the case that no position holds its own number?