Dummit and Foote's exercise 14.3.11 asks to prove that $f(x) = x^{p^{n}}-x+1$ is irreducible over $\mathbb{F}_{p}$ iff $n=1$ or $n=p=2$. To prove the 'only if' part, the exercise suggest to prove that if $\alpha$ is a root of $f(x)$, then so is $\alpha + a$ for every $a \in \mathbb{F}_{p^{n}}$. This implies that $\mathbb{F}_{p}(\alpha)$ contains $\mathbb{F}_{p^{n}}$ and that $[\mathbb{F}_{p}(\alpha):\mathbb{F}_{p^{n}}] = p$.   I've proven everything except that $[\mathbb{F}_{p}(\alpha):\mathbb{F}_{p^{n}}] = p$. Can anyone give me some ideas for this? Thanks.
I have a question, I think it concerns with field theory.     Why the polynomial $$x^{p^n}-x+1$$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$?   Thanks in advance. It bothers me for several days.
What software do you use to accurately draw geometry diagrams? How do I set Windows to boot as the default in the boot loader? Is there an easy way to put a box around a theorem in LaTeX? For example to state an important theorem. I tried using page 20 of the ntheorem , but I do not know how to use the package. I'm trying to write a blog post which includes a code segment inside a &lt;pre&gt; tag. The code segment includes a generic type and uses &lt;&gt; to define that type. This is what the segment looks like:  &lt;pre&gt;     PrimeCalc calc = new PrimeCalc();     Func&lt;int, int&gt; del = calc.GetNextPrime; &lt;/pre&gt;   The resulting HTML removes the &lt;&gt; and ends up like this:  PrimeCalc calc = new PrimeCalc(); Func del = calc.GetNextPrime;   How do I escape the &lt;&gt; so they show up in the HTML? Google Chrome will take up my memory to the point where it causes my computer to freeze to a near halt. What can I do to prevent this? It gets so horrible that I can not even switch to TTY1 to force a reboot. I have to hard reset, and I lose work.  Ideally, I would like to be able to use the chrome browser on Ubuntu without having to keep reminding myself to close tabs.   Are there any extensions for Chrome, or tools for Ubuntu that can help prevent me going past the point where Ubuntu eats up all the RAM and becomes incapacitated? Looking for the title of a gold heist movie involving the telekinetic son of an extraterrestrial/alien? Some 30 years back, I saw a film about the son of an extraterrestrial who gets involved in a gold heist.  The main character does not initially know that he is the son of an extraterrestrial and that has special telekinetic powers. The film opens, I think, with him being in some train yard, his foot getting stuck in a switch with a locomotive approaching. He stops it using his powers intuitively, possibly passing out (not realizing what he actually did). I think his mother subsequently reveals his heritage to him.  He somehow gets hired as a bodyguard for someone planning a gold heist. I think they know his abilities and use him to steal a million US dollars. Actually, that is just a distraction, covering that they steal 30 million dollars from the same location (through the air, not sure on that part).  The last part I remember is some underground tunnels where the thieves set up to melt the gold. The main character is locked in there.  Does anyone know the title of this movie? Searching for things like "sci-fi gold heist extraterrestrial etc." did not reveal anything to me. I also had a hard time picking proper tags (sorry). Story about a deadly alien creature on another planet that has some chemical in its body that makes it faster I can not remember if it was a short story or a novel, but I am leaning towards it being a novel. There is a colony of humans and I can't remember if they crash-landed or not.   There is a creature that they come to discover after it has killed a few people. The creature has some kind of chemical in its body that is like a super-adrenaline. One part is I remember is one of these creatures getting very hot from using up its reserve of this chemical and it dived into a river or pond and it started to steam. Something like that.  It has to be at least 10 years since I read the story. $\sqrt{1-x^2}$ is not a rational function If $k$ is a field in which $1 + 1 \neq 0$, prove that $\sqrt{1-x^2}$ is not a rational function. Hint. Mimic the classical proof that $\sqrt{2}$ is irrational   It seems rather odd to discuss the square root function in the context of an arbitrary field, but here goes nothing!  Proof:   Suppose that $\sqrt{1-x^2}$ is a rational function in $k(x)$. Then there exist polynomials $p(x)$ and $q(x) \neq 0$ that are relatively prime with $\sqrt{1-x^2} = \frac{p(x)}{q(x)}$; clearly we may take $p(x) \neq 0$. Then $1-x^2 = \frac{p(x)^2}{q(x)^2}$ or $q(x)^2(1-x^2) = p(x)^2$, which says $q^2 | p^2$, and since $q | q^2$, we can infer $q | p^2$. Moreover, since $p|p^2$ and $(p,q)=1$, then $pq|p^2$ or $q|p$, which contradicts the fact that the polynomials are relatively prime. Hence $\sqrt{1-x^2}$ cannot be a rational function.    How does this sound? Aside from the problem of discussing $\sqrt{~~}$ in the context of an arbitrary field, I am worried about not using the fact that $1+1 \neq 0$, at least not explicitly. Where exactly is this assumed used, if at all?  EDIT:  Suppose that $q(x) =1$, and let $f(x) = a_n x^n + ... a_1 x + a_0$. Then $1-x^2 = f(x)^2$. If $x=0$, $1 = a^2_0$ and therefore $a_0$ must be a unit. Letting $x=1$ we get $0 = (a_n + ... + a_1 + a_0)^2$; and letting $x=-1$ we get $0 = (-a_n - ... - a_1  + a_0)^2$. Since we are working in a field, there can be no nonzero nilpotent which means that $a_n + ... + a_1 + a_0$ and $-a_n - ... - a_1  + a_0$ are both zero. Adding the two equations together yields $2a_0 = 0$, and since $1+1 \neq 0$, $a_0 = 0$ which contradicts the fact that $a_0$ is a unit.   How does this sound?  Another attemtpt:  If $f(x)^2 = 1-x^2$, then $2 = \deg(f^2) = 2 \deg (f)$ implies $\deg(f) = 1$. Yet if $x=-1$, we get $f(-1)^2 = 1 - 1 = 0$ and therefore $f(-1)=0$ since there are no nonzero nilpotent elements in a field; similarly, $f(1)=0$. Since $1+1 \neq 0$, $f$ has two distinct roots in $k$ yet is only a $1$-st degree polynomial, a contradiction. When to choose checked and unchecked exceptions In Java (or any other language with checked exceptions), when creating your own exception class, how do you decide whether it should be checked or unchecked?  My instinct is to say that a checked exception would be called for in cases where the caller might be able to recover in some productive way, where as an unchecked exception would be more for unrecoverable cases, but I'd be interested in other's thoughts. Towards the end of , how do I get  $$ \frac{\sqrt{8z + 1} - 1}{2} &lt; w + 1 $$  Here $w = x + y \geq 0, t = (w^2 + w)/2 , z = t + w, x,y \in \mathbb N_{\geq 0}$.  Thanks. Optimizing logical sector size for physical sector size 4096 HDD Why is setTimeout(fn, 0) sometimes useful? Concrete Mathematics - The Josephus Problem I've been going through  and have a question on the inductive proof for the . Recurrent relation:  $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + 1$  Closed form hypothesis:  $J(2^m + l) = 2l + 1$, for $m \ge 0$ and $0\le l&lt;2^m$  Inductive Proof (for the even case):  $J(1)=1$ $J(2^m+l)= 2J(2^{m-1}+l/2)-1=2(2l/2+1)-1=2l+1$  How do the authors get from $2J(2^{m-1}+l/2)-1$ to $2(2l/2+1)-1$? $A_n$ is the only subgroup of $S_n$ of index $2$. How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$?  Why isn't there other possibility?  Thanks :) What is the strict aliasing rule? Connecting two Ethernet PHY without magnetics? In a (tentative) new design I want to connect two 100 Mbit/s Ethernet PHY similar to , separated by few inches on the same PCB, with same ground plane (but different power supplies). I have the choice of one of my PHY, but the other is embedded in an as-yet-unspecified PCIe to Ethernet IC (perhaps Gigabit but used in 100 Mbit/s mode), and it is critical that this PHY thinks there is a bona fide 100 Mbit/s Ethernet connection.  I could use the following         &ndash; Schematic created using   but I would like to achieve the same without the magnetics, in order to save on cost, purchasing headaches, and if at all possible power consumption.  I can imagine an RC network, perhaps as simple as in this , found thanks to a ; or maybe engineered with attenuation and lowpass, but that will not emulate the fact that with real magnetics, when there is a negative pulse on TX1P, there is a positive pulse on TX1M and RX2M. I am uncertain about if that would prevent some PHYs from working normally.  Anything to suggest that one is confident will work for practically any PHY around? How can a capacitive touch screen be triggered without human contact? I want to robotically touch an iPhone screen without any human intervention.   I've done some experiments but have not found a reliable solution. It seems even styluses that work through gloves still rely on the capacitive characteristics of the human body.    Carrot held by human works Carrot held in plastic clamp doesn't Pogo Stylus works held by human and held by human through clothes Stylus does not work held in plastic clamp.  Stylus works held in plastic clamp attached to human through clothes via jump lead. Stylus held in plastic clamp with jump lead dangling off seems to work most the time.   Is it possible to mimic the human element using capacitors and other components? What signal does the capacitive screen need to be triggered?  Thanks Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$.  Form the Cartesian Product of these sets $X=\displaystyle\prod_{n=1}^{\infty}X_n$, and define $\rho:X\times X\rightarrow\mathbb{R}$ by  $$\rho(x,y)=\displaystyle\sum_{n=1}^{\infty}\frac{\rho_n(x_n,y_n)}{2^n[1+\rho_n(x_n,y_n)]}.$$  Show that $\rho$ is a metric on $X$ whose induced topology is equivalent to the product topology on $X$.  So basically what this problem is saying is that there's a canonical way to define a metric on the countable product of metric spaces.  I showed in a previous problem that the topology induced by $\rho_n$ is equivalent to that induced by $\frac{\rho_n(x_n,y_n)}{1+\rho_n(x_n,y_n)}$.  And thus we can go ahead and just assume that $\rho_n&lt; 1$ for all $n$ and replace our infinite series by  $$\rho(x,y)=\displaystyle\sum_{n=1}^{\infty}\frac{\rho_n(x_n,y_n)}{2^n}.$$  Now comes the interesting part: how should I go about showing that the product topology on $X$ and the topology induced by $\rho$ are equivalent?    The basis for the product topology given to me in my book's definition is that of cartesian products made up the $X_n$ except for finitely many which are $O_n$ for some open subset of $X_n$.  However I believe I was able to improve upon this and show that I could decompose these into a basis where the $O_n$ were all open balls induced by their respect $\rho_n$ metric.  For $\rho$ I'm using the basis of open balls that it induces, as I can see no other reasonable choice.    However I can't seem to match these two bases up.  There are many different points $\{x_n\}\in X$ which make my infinite series less than a certain value and there is so much freedom in which terms in the series I choose to reduce in size that it seems hopeless to try and fit an open ball induced by $\rho$ into any one of my basis elements for the product topology.    Is there a more appropriate strategy for proving these two topologies are equivalent? "I will appreciate your help" I encounter the phrase "I will appreciate your help" from time to time and it doesn't sound too honest to me. To be more specific, I don't have a problem with this phrase coming from a friend, because he will appreciate my help by "owing me one". But a stranger from the Internet, that's different, methinks.  "How will you appreciate it?" pops in my mind every time I see it, because to me, appreciation equals reward and, well, you can't reward people over the internet much. It is quite possible that I am just wrong and this is a perfectly fine figure of speech. No more WIFI. oem-wifi-intel-iwlwifi doesn't build on the new 4.4 kernel