Show that the polynomial $(x-1)(x-2)...(x-n)-1$ is irreducible over $\mathbb{Z}$. [hint: if the polynomial factors consider the values of the factors at $x = 1,2,3....,n$.  Show that the polynomial $(x-1)(x-2)...(x-n)+1$ is irreducible over $\mathbb{Z}$ $\forall n \geq 1, n \neq 4$.  I've been running around in circles on this one for awhile now and can't seem to get anywhere! Some insights would be greatly appreciated! Thanks yall!
I try to solve this problem. I seems to come close to the end but I can't get the conclusion. Can someone help me complete my proof. Thanks          Show that the polynomial $h(x) = (x-1)(x-2)\cdots(x-n) + 1$ is irreducible over $\mathbb Z$ for all $n\ge1$ and $ n\ne4$.      Suppose $h(x) = f(x) g(x)$, then we must have $f(i)g(i) = 1$ for all $i = 1,2,...n$. So both $f(i)$ and $g(i)$ are $1$ or $-1$. In either case, $m(x) = f(x) - g(x)$ has degree smaller than $n$ and have $n$ different roots ($1,2,...,n$). So we must have $m(x) = 0$. Then $h(x) = f(x)^{2}$. So $n$ must be even. Let $n = 2k$. Because $f(x)$ has degree $k$, there are $k$ values from $\{1,2,...,2k\}$ at which $f(x)$ is $1$ and $k$ values at which $f(x)$ is $-1$.  This is where I got stuck. Hope some one can help me solve this. Thanks.
Extracting only the BibTex records relevant to the paper My colleague asked me to write a page for a paper he writes. I wrote the page in LyX, compiled it to LaTeX, and sent him the TeX files. Then he reminded me that I forgot to send the .bib file with all the references.  But, my .bib file is huge - it contains over 1000 references (I have a global BiBTeX file for all my papers). The page I wrote uses only about 20 of them.  Is there an automatic way to extract from the bib file, only the records relevant to the current paper? To show: $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$  Now, the equation $x^{2} =3$ has a solution in $\mathbb{R}$, but not in $\mathbb{Q}$.  Hence they are not isomorphic to each other.  Is that right, or do I need to prove something else?  Thanks. What does a confidence interval with a negative endpoint mean? Expressing the area of the cross section of a cylinder I was able to solve similar questions where a triangle was involved, but this one has had me looping and pondering on it for the past week.  The Condition: Let $r$ be a positive constant. Consider the cylinder $x^2+y^2\leq r^2$ and let C be  part of the cylinder that satisfies $0\leq z\leq y$.  The questions:  (1) Consider the cross section of $C$ by the plane $x=t(-r\leq t\leq r)$ and express its area in terms of $r,t$  (2) Calculate the volume of $C$, and express it in terms of $r$.  (3) Let $a$ be the length of the arc along the base of the circle $C$ from the point $(r,0,0)$ to the point $(r cos\theta,r sin\theta,0)(0\leq\theta\leq\pi)$. Let $b$ be the length of the line segment from the point $(r cos\theta, r sin\theta, 0)$ to the point $(r cos\theta, r sin\theta, r sin\theta)$. Express $a$ and $b$ interms of $r,\theta$  (4)Calculate the are of the side of $C$ with $x^2+y^2=r^2$, and express it in terms of $r$.  Here are the expected answers(which I keep failing to get to):  (1) $\frac{1}{2}(r^2-t^2)$  (2) $\frac{2}{3}r^3$  (3) $r\theta$ and $r sin\theta$  (4) $2r^2$ How to sync gnome-calendar with an online calendar? I thought it was awesome that Ubuntu 16.04 shipped with gnome-calendar, but I noticed that holidays were not included.  I also noticed that there is an option to sync the calendar with an online one, how do I do this?  I don't know of any online calendars I could use. Grandi's Series; tends to $1/2$, but why is this considered a valid sum? For example, if I revisit a memory of mine where I'm inside a castle and talking to someone, can the present-me explore different parts of the castle meanwhile the past-me continues her conversation? Is that possible? Updating the SubscriberKey on the All Subscriber table through the API I'm trying to update an existing Subscriber in the All Subscriber Table through the API; i've set up the AppId in the App Center and have connected and created Subscribers, but i'm still having issues updating.  What call do I need to be making to do this? These are the ones i've been trying. What are all possible ways to stop a zombie from despawning? Is there any way for which we are able to stop a zombie from de-spawning no matter how far we go? If so, what are all possible ways?  Bonus question: what are all the other techniques for which we can prevent a zombie from de-spawning within a specific block radius? Let $U$ be connected. A function $u \in H^1(U)$ is a weak solution   of Neumann's problem   \begin{equation} (*)\qquad\left\{ \begin{array}{rl} -\Delta = f &amp; \text{in } U \\ \frac{\partial u}{\partial \nu} = 0 &amp; \text{on } \partial U \end{array} \right. \end{equation}   if   $$ \int_U Du \cdot Dv \; dx = \int_Ufv \; dx $$   for all $v \in H^1(U)$. Let $f\in L^2(U)$. Prove $(*)$ has a weak solution   if and only if   $$ \int_U f \; dx =0. $$   For the only if part I set $v=1$. However I do not see where to start the if part. I was thinking of the Lax-Milgram theorem. That's where I'm now. I have a question about calculating the degree of a finite field extension over $\mathbb{Q}$.  This is problem 18 in chapter 1 of Patrick Morandi's Field and Galois Theory. The problem asked to show that $$ \left[ \mathbb{Q}{ \left( \sqrt[4]{2},\sqrt{3} \right):\mathbb{Q} }\right] =8 $$ Here's my work : First, we have $$  \left[ \mathbb{Q} ( \sqrt[4]{2},\sqrt{3}):\mathbb{Q}\right] = \left[ \mathbb{Q}(\sqrt[4]{2},\sqrt{3}) : \mathbb{Q}({\sqrt[4]{2}}) \right] . \left[ \mathbb{Q}{(\sqrt[4]{2})}: \mathbb{Q} \right]$$ It's easy to show that $\left[ \mathbb{Q}{(\sqrt[4]{2})}: \mathbb{Q} \right] =4 $ because the minimal polynomial of $\sqrt[4]{2}$ over $\mathbb{Q}$ is $x^4 -2$. Next, I'm going to show that  $$ \left[ \mathbb{Q}(\sqrt[4]{2},\sqrt{3}) : \mathbb{Q}({\sqrt[4]{2}}) \right] =2$$ Consider the polynomial $f(x) = x^2 -3$. It's monic and $\sqrt{3}$ is a root of $f$, but I'm stuck in proving $\sqrt{3} \notin \mathbb{Q}(\sqrt[4]{2})$ (hence $f$ is irreducible over $\mathbb{Q}(\sqrt[4]{2}))$.  My question is: Is there any proof to show that $\sqrt{3} \notin \mathbb{Q}(\sqrt[4]{2})$ or another way to caculate the degree of this field extension?  Thank you for your help. Which infinity is meant in limits? For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why? Word with the same meaning as "good moments" or "pleasant memories"? Is there a single word with the same meaning as "good moments" or "pleasant memories"? I'm looking for a word which describes good thing(s) that happened to someone. The only word I can think of is "joy".  Some examples might be:   Wayne has many pleasant memories of the days spent with the family. She could remember many good moments from the last weekend. I'm a mathematician interested in abstract QFT.  I'm trying to undersand why, under certain (all?) circumstances, we must have $T^2 = -1$ rather than $T^2 = +1$, where $T$ is the time reversal operator.  I understand from the Wikipedia article that requiring that energy stay positive forces $T$ to be represented by an anti-unitary operator.  But I don't see how this forces $T^2=-1$.  (Or maybe it doesn't force it, it merely allows it?)  Here's another version of my question.  There are two distinct double covers of the Lie group $O(n)$ which restrict to the familiar $Spin(n)\to SO(n)$ cover on $SO(n)$; they are called $Pin_+(n)$ and $Pin_-(n)$.  If $R\in O(n)$ is a reflection and $\tilde{R}\in Pin_\pm(n)$ covers $R$, then $\tilde{R}^2 = \pm 1$.  So saying that $T^2=-1$ means we are in $Pin_-$ rather than $Pin_+$.  (I'm assuming Euclidean signature here.)  My question (version 2): Under what circumstances are we forced to use $Pin_-$ rather than $Pin_+$ here?  (I posted a  on physics.stackexchange.com last week, but there were no replies.)    EDIT: Thanks to the half-integer spin hint in the comments below, I was able to do a more effective web search.  If I understand correctly, Kramer's theorem says that for even-dimensional (half integer spin) representations of the Spin group, $T$ must satisfy $T^2=-1$, while for the odd-dimensional representations (integer spin), we have $T^2=1$.  I guess at this point it becomes a straightforward question in representation theory: Given an irreducible representation of $Spin(n)$, we can ask whether it is possible to extend it to $Pin_-(n)$ (or $Pin_+(n)$) so that the lifted reflections $\tilde R$ (e.g. $T$) act as an anti-unitary operator. Ambiguity of “not.... because” You are not hired because you are qualified.   I think this means you are hired but it is not because you are qualified. But I think this can  also means you are not hired, and the reason for it is that you are qualified.   Does this ambiguity really exist or do I mistake anything? Prove Uncountable set minus a countable set is uncountable Let B be a countable set, and let A be an uncountable set. Prove that A-B is uncountable How to call a SOAP web service on Android I am having a lot of trouble finding good information on how to call a standard SOAP/WSDL web service with Android. All I've been able to find are either very convoluted documents and references to "kSoap2" and then some bit about parsing it all manually with . OK, that's fine, but it's 2008, so I figured there should be some good library for calling standard web services.  The web service is just basically one created in . I would like to have IDE support for generating the plumbing classes. I just need the easiest/most-elegant way to contact a  based web service from an Android-based phone. Remove a child with a specific attribute, in SimpleXML for PHP I have several identical elements with different attributes that I'm accessing with SimpleXML:  &lt;data&gt;     &lt;seg id="A1"/&gt;     &lt;seg id="A5"/&gt;     &lt;seg id="A12"/&gt;     &lt;seg id="A29"/&gt;     &lt;seg id="A30"/&gt; &lt;/data&gt;   I need to remove a specific seg element, with an id of "A12", how can I do this?  I've tried looping through the seg elements and unsetting the specific one, but this doesn't work, the elements remain.  foreach($doc-&gt;seg as $seg) {     if($seg['id'] == 'A12')     {         unset($seg);     } } Filter the score of the questions on the homepage before loading them At the moment questions on the homepage are loaded by the top N most recent and then ones with a score of -4 or less are filtered out. This doesn't make much difference for a user who is logged in but consider anonymous users; their home page is only 15 questions in length. If someone was to edit several posts that bump to the top with a score of -4 or less then the home page looks tiny. At the moment on SFF, for example, the homepage for anonymous users only shows 9 questions:  At one point it only showed 3 questions (a few weeks back):  Can the filtering logic be changed to before the questions are loaded to appear on the homepage? PhD without previous academic degree: truth or myth?