I have already shown (in a previous question) that $I=\langle x,2\rangle$, and that I is a prime ideal, and I believe that allows me to show that $\mathbb{Z}[x]/I$ is an integral domain, which if it is finite would allow me to conclude that $\mathbb{Z}[x]/I$ is a field, which would imply that I is a maximal ideal. But I don't know how to determine the number of elements in $\mathbb{Z}[x]/I$, and thus do not know how to determine whether or not it is finite.   Can someone tell me if this thinking is correct, and if so how to determine the number of elements in $\mathbb{Z}[x]/I$?   Note: we have not covered irreducibility in this course.
In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$   In fact, $I=\left \langle x,2 \right \rangle$  How do I show that the order of $\mathbb{Z}\left [ x \right ]/I$ is 2?
What should children call a step-grandparent? My mom remarried to a great guy who has been absolutely wonderful to my kids. Ever since my mom and he were dating my kids have called him Mr. Joe. Lately my mom has been bugging me to get the kids to call him Grampa.  I agree that he deserves more respect than Mr.  The problem is that he isn't their real grampa so I don't really want them calling him that. Does anybody have any suggestions on what we should have the kids call him? They are 9 and 12 years old.  EDIT: I appreciate the comments regarding different variations of grampa and hearing how others have no issues with calling their step-granddad grampa. But I was hoping maybe someone has some suggestions for names that show the importance of the step-granddad without calling him some variation of the grampa name but still has a feeling of warmth and family in the name (I hope that makes sense). What does '>/dev/null 2>&1' mean in this article of crontab basics? Show that if $a \neq b$ and a and b are positive then $\frac{a}{b}+\frac{b}{a}$ is never an integer When I create beamer frame, the content is centred in the slide. How do I make the content move to the top of the slide by default? Uncaught rerender threw an error in 'markup://aura:expression' [info.$component$.$isValid$ is not a function] i'm trying to set a list of string to an aura:attribute and i get this error:     Uncaught rerender threw an error in 'markup://aura:expression'   [info.$component$.$isValid$ is not a function]   that's the attribute:  &lt;aura:attribute name="parents" type="String[]"/&gt;   i call this function in js:  test: function(component){     var action = component.get("c.test");         //Set up the callback         var self = this;         action.setCallback(this, function(actionResult) {                   debugger;             component.set("v.parents", actionResult.getReturnValue());         });                 $A.enqueueAction(action);    }   and thats the function:  @AuraEnabled public static List&lt;String&gt; test(){     String[] a = new String[]{};     a.add('1');     a.add('2');     return a; }   please help Thanks Disease transmitted by postage stamps Sums in math mode: how to display index under the sigma sign? I have a fraction with sums above and under the line. How can I convince LaTeX to write the indices of the sums under the sigma instead of next to it?  \begin{displaymath}   \frac{\sum_{s \in S} s^2}{\sum_{p \in P} p^2} \end{displaymath} How to call shell commands from Ruby How do I call shell commands from inside of a Ruby program? How do I then get output from these commands back into Ruby? Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits. Recommendation for mobile prepaid SIM card for smartphone + home calls in Germany What SIM card would you recommend for a smartphone user for travel in Germany?  It would be probably the only or the most important connection to the Internet, so large amounts of data and relatively fast transfer is an issue.  I need to stay in touch with friends and family, so the calls to other EU countries should be as cheap as possible.   For to the first criteria, I found the T-Mobile offer interesting. The internet costs 10 Euros/month and after reaching the transfer limit it stays relatively fast. According to tests (Galileo from Pro7, AFAIR) it has also the best coverage. However, calls to other countries are very expensive.  For to second criteria, Lebara and Mobilka are interesting. The calls are very cheap, but there are problems with signal strength in many locations, where T-Mobile has no problem. The voice quality, at least in Lebara, is also limited. They both have mobile internet with unlimited transfer, but after reaching the high speed limit, the transfer is reduced to 64 kbps, which is too slow.  I've found out that O2 has foreign calls at moderate prices, but I haven't yet checked for details. There are so many other options, making it is hard to compare them all without spending the whole weekend on it. So, if someone has found a better option than having 2 SIM cards and constantly switching them, this answer would be greatly appreciated. I have got this problem on limits  $$ \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$$ What I am doing here is that taking log and then applying L-Hospital's Rule $$y= \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2} $$ then $$\ln(y)=\lim_{x \rightarrow 0} \frac{1}{x^2}\ln\left(\frac{\sin(x)}{x}\right)$$ now it becomes $\frac{0}{0}$ type , but problem is how to take derivative here? when differentiating $\ln\left(\frac{\sin(x)}{x}\right)$ it will become $\frac{x}{\sin(x)}$ then we'll have to differentiate one more time like we do in case of $f(f(x))$ ? $\sin(x)$ and $x$ will be separately differentiated or together ? A commercial airport being any airport that has regular scheduled passenger flights.  The mathematics of  got me thinking: is the network of scheduled flights a connected graph?  That is, is there a pair (or triplet or even larger set) of airports such that they only have scheduled flights with each other?  Or can you get from any airport to any other?  Ignore vagaries like cargo and general aviation. I'm looking for two airports that I can shuttle between, but would have to drive (or sail) to get to one or the other of them. preventDefault() on an  tag DES-X , computation load and storage If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative. Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$. Prove that $R$ must be a commutative ring. Internet through wireless hotspot, intranet through wired network - can I use both at the same time? I have a module with a page callback handler for the path "products". So if the user navigates to products/apples then the module presents a page about our apples, and if they go to products/oranges then etc. These pages include detail about the various products.  I would like XML Sitemap to be populated from my module. After all, it's my module that knows what products exist, their descriptions etc.  Any hook or anything I can use for that? Alright Ubuntu wizards, I jacked up my system a bit unnecessarily, which is now giving me the chance to learn more about how to fix it...a vicious cycle. Anyway, I have it mostly up and running again to my liking, though I am receiving this (photo attached) prompt when my system boots:  libkmod: ERROR ../libkmod/libkmod-config.c:686 kmod_config_parse: /etc/modprobe.d/blacklist.conf line 56: ignoring bad line starting with 'rt2800usb'   Other users had success with just deleting the bad line from their blacklist.conf file, which I have done. Line 56 no longer exists, but I'm still receiving the same error. So am I missing something? Is there some follow-up command I need to give Ubuntu after saving the conf file? Want \section* to change the page header In a book, I have several numbered sections followed by an unnumbered section. I have placed an \addcontentsline following the \section* that introduces the unnumbered section, so that it will appear (unnumbered) in the ToC; that works fine. However, the page headers for the unnumbered section are continued from the last numbered section.  I am using fancyhdr with the following setup:  \pagestyle{fancy} \fancyheadoffset{0in} \renewcommand{\chaptermark}[1]{\markboth{#1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}  \fancyhf{} \fancyhead[LE,RO]{\thepage} \fancyhead[C]{\bfseries \draftinfo} \fancyhead[LO]{\sf\rightmark} \fancyhead[RE]{\sf\leftmark} \fancyfoot[C]{\bfseries \draftinfo} \fancypagestyle{plain}{%   \fancyhead{} % get rid of headers   \renewcommand{\headrulewidth}{0pt} % and the line } Reverse of number in PHP not showing correct output