Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11 Let $K=k(x)$ be the rational function field over a perfect field $k$ of  characteristic $p &gt; 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and  $g$ relatively prime. Show that $K$ is  separable over $F$ if and only if $u$ is not in $K^p$.
Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$. Let $K$ be the rational function field $k(x)$ over a perfect field $k$ of   characteristic $p &gt; 0$. Let $F = k(u)$ for some $u\in K$, and write   $u = f(x)/g(x)$ with $f$ and $g$ relatively prime. Show that $K/F$ is   a separable extension if and only if $u\notin K^p$.   There is already a post on this but it has not yet been answered :   The truth is that I am very confused with this problem, I would like someone please help me to understand what happens with this exercise and explain to me how it could be done by giving me a help or something. I already have an approach for the left-to-right implication, but I do not know very well how good it is. Could someone help me with the other implication please? Thank you very much.
Hey all I downloaded this file () but can't seem to be able to uninstall it. I tried looking online as well as the README file but there is only instruction for installing, not uninstalling the package.   Attempted solutions:  sudo apt-get remove laptop_mode   returned   E: Unable to locate package laptop_mode   but when I type in laptop_mode I do get   Unhandled kernel version: 3.13 ('uname -r' = '3.13.0-32-generic')   Is there any way I can uninstall laptop-mode-tools? I've only been able to resolve this far  $$ d \mid n \Rightarrow \exists l \in \mathbb{Z}\; \textrm{such that} \; n=dl \Rightarrow x^{q^n} - x = x^{q^{dl}} - x.$$  I can't move forward. Profile picture cannot have transparent background How to add a Skype indicator? How can I blow out the background when shooting down for clothing on the ground? This is a  about RAID levels.    What are:   the RAID levels typically used (including the RAID-Z family)?  deployments are they commonly found in? benefits and pitfalls of each? Is there any way to automatically unwrap UV map proportionally to object or "world" size?  For example:  I've made two cubes, one is two times bigger than other (scale is applied). Then I unwrapped texture with Smart UV project which will try to span all faces on whole texture space(0 to 1) so bricks on left cube will be bigger than bricks on right cube.   But what I want to achieve is this:     In broader context:  When I import model from Archicad (in form of OBJ) all objects have UV's in some scale (so all bricks have the same size on every imported object). And when I want to create custom object in scene I have to manually scale UV's to adjust bricks size for every new object (instead of scaling to one constant parameter). What is the origin of non-conservative force? My understanding about  is a force that its work is independent of path such that we can construct another form of the work called potential to make our life easier.  For friction, if I start from microscopic point of view, it should be the macroscopic effect of the electric force or gravity which are both conservative force.  Why do we initially have description by conservative forces (electric/gravity force) but end up with a macroscopic description of nonconservative force(friction)? how to run a winform from console application? How do I create, execute and control a winform from within a console application? Google chrome is reading characters backwards Google Chrome is my favorite browser, and I can't imagine going on without it. Suddenly (without closing and re-opening) the browser; chrome has been reading characters upside down, very weird:    Chrome version:    I tried restarting my browser and my MAC (running OS X Yosemite 10.10), but the error keeps on coming. Can a .NET windows application be compressed into a single .exe? Automatically redirect anonymous user from unanswered duplicate question to corresponding answered version Refreshing a question doesn't update the answer count properly In the Stack Exchange Android App v1.0.7 1.0.33 (with a HTC One running Android 4.4.2), the answer count for a question you're viewing is displayed in two places:   In the label of the answer tab:   At the bottom of the question tab:    I've noticed that these counts don't reliably update when new answers arrive.   If you refresh the question tab, neither count updates. If you refresh the answer tab, the second count will be updated, but the answer tab's count will not, leaving the two out of sync.   Refreshing either tab should probably update both counts!    Screenshots of the bug in action:    I waited for it to be answered. A while later, the question now has an answer.    The tab still says 0 answers though. Switching back to the question tab, , but the tab count still doesn't. Probably a novice question, but how do I correctly diagnose from a photo for what reason that camera wasn't able to focus properly?  It can be any 1 of the following   Unsteady hand Lens issue Metering Re-Calibration required in camera or ...   Is there a set of unsaid protocols that needs to be followed by a photographer?  Goal is to get a sharp and crisp image prior to editing.  [PS: I own a Nikon D5200.] Unique pair of positive integers $(p,n)$ satisfying $p^3-p=n^7-n^3$ where $p$ is prime Q. Find all pairs $(p,n)$ of positive integers where $p$ is prime and $p^3-p=n^7-n^3$.   Rewriting the given equation as $p(p+1)(p-1)=n^3(n^2+1)(n+1)(n-1)$, we see that $p$ must divide one of the factors $n,n+1,n-1,n^2+1$ on the $\text{r.h.s}$.   Now, the $\text{l.h.s}$ is an increasing function of $p$ for $p\ge1$. This implies that for any given $n\ge1$, there is exactly one real $p$ for which $\text{l.h.s}=\text{r.h.s}$. For $p=n^2$, we get $\text{l.h.s}=n^6-n^2&lt;n^7-n^3=\text{r.h.s}.$ This means that either $p&gt;n^2$ or $p&lt;n^2$ must hold.  Assuming $p&gt;n^2$, it follows that the prime $p$ cannot divide any of $n,n+1,n-1$. So $p$ must divide $n^2+1$ and hence $p=n^2+1\quad (\because p&gt;n^2)$.  Substituting the value of $p$ in the given equation we get, $n^2+2=n^3-n\implies n^3-n^2-n=2$. As the factor $n$ on the $\text{l.h.s}$ must divide $2$, the above equation has a unique integer solution $n=2$.  Finally, we get $(5,2)$ as the solution to the given equation.      But how do I conclude this is the only solution possible? Also, why does'nt $p&lt;n^2$ (the case which I ignored) hold? As a bonus question, I would like to ask for any alternative/elegant solution (possibly using congruence relations) to the problem. What is the best collation to use for MySQL with PHP? Intuitive examples of importance sampling My background is computer science. I am fairly new to monte carlo sampling methods and, although I understand the math, I have hard time coming up with intuitive examples for importance sampling. More precisely, could someone provide examples of:    an original distribution one cannot sample from but one can estimate an importance distribution which can be sampled from and adequate for this original distribution. How do I assign the output of a command to a variable? I often read that in case of Deep Learning models the usual practice is to apply mini batches (generally a small one, 32/64) over several training epochs. I cannot really fathom the reason behind this.  Unless I'm mistaken, the batch size is the number of training instances let seen by the model during a training iteration; and epoch is a full turn when each of the training instances have been seen by the model. If so, I cannot see the advantage of iterate over an almost insignificant subset of the training instances several times in contrast with applying a "max batch" by expose all the available training instances in each turn to the model (assuming, of course, enough the memory). What is the advantage of this approach? I updated some entries in the  at C:\WINDOWS\System32\drivers\etc.  As a test, I decided to make Google point to 127.0.0.1.  # Additionally, comments (such as these) may be inserted on individual # lines or following the machine name denoted by a '#' symbol. # # For example: # #      102.54.94.97     rhino.acme.com          # source server #       38.25.63.10     x.acme.com              # x client host  127.0.0.1       localhost ::1             localhost 127.0.0.1       google.com   This works fine on Windows Vista, but not on Widows 7. When I type google.com into my browser, it goes directly to Google's website.  I am not using a proxy server, and I think there might be some temporary DNS settings that need to be flushed, but I'm not sure.  Does anyone know how to fix this?