Let $R=\mathbb{Z}[i]/\langle2-2i\rangle$.  I want to know the number of elements in $R$.  The function $N$ given by $N(a+bi)= a^2+b^2$ is the norm, so $N(a+bi) &lt; N(2-2i)$.  This leaves us with representatives $$ \{0,1,-1,i,-i,1+i,1-i,-1+i,-1-i,2,-2,2i,-2i\} \, . $$ So in this particular factor ring, $2 = 2i$. Then squaring both sides we see $1 = i$, so $R = \{I,1+I,-1-I, 2+I,-2+I\}$, where $I=\langle2-2i\rangle$.  So I find that the number of elements of $R$ is five. Is this solution correct?
The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is , when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d&gt;1$?      — For instance if $p$ is prime (which is not coprime with $0$) then $$\mathbb{Z}[i]/(p) \cong \mathbb{F}_p[X]/(X^2+1) \cong \begin{cases} \mathbb{F}_{p^2} &amp;\text{if } p \equiv 3 \pmod 4\\ \mathbb{F}_{p} \times \mathbb{F}_{p} &amp;\text{if } p \equiv 1 \pmod 4 \end{cases}$$ (because $-1$ is a square mod $p$ iff $(-1)^{(p-1)/2}=1$).  — More generally, if $n=p_1^{r_1} \cdots p_m^{r_m} \in \Bbb N$, then  each pair of integers $p_j^{r_j}$ are coprime, so that by CRT we get $$\mathbb{Z}[i]/(n) \cong \mathbb{Z}[i]/(p_1^{r_1}) \times \cdots \times \mathbb{Z}[i]/(p_m^{r_m})$$  I was not sure how to find the structure of $\mathbb{Z}[i]/(p^{r}) \cong (\Bbb Z/p^r \Bbb Z)[X] \,/\, (X^2+1)$ when $p$ is prime and $r&gt;1$.  — Even more generally, in order to determine the structure of $\mathbb{Z}[i]/(a+bi)$ with $a+bi=d(x+iy)$ and $(x,y)=1$, we could try to use the CRT, provided that $d$ is coprime with $x+iy$ in $\Bbb Z[i]$. But this is not always true: for $d=13$ and $x+iy=2+3i$, we can't find Gauss integers $u$ and $v$ such that $du + (x+iy)v=1$, because this would mean that $(2+3i)[(2-3i)u+v]=1$, i.e. $2+3i$ is a unit in $\Bbb Z[i]$ which is not because its norm is $13 \neq ±1$.  — I was not able to go further. I recall that my general question is to known what $\mathbb{Z}[i]/(a+bi)$ is isomorphic to, when $a$ and $b$ are integers which are not coprime (for instance $a=p^r,b=0$ or $d=(a,b) = a^2+b^2&gt;1$).  Thank you for your help!
One of is followed by plural noun/pronoun and singular helping verb How do I choose an element from a non-empty set? Suppose I have a non-empty set $A$.     How do I choose an element $x\in A$?   More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is exactly one $x\in y$ such that $P(x,y)$ is true.  I have thought about this for a while without success and I am beginning to doubt such a formula is even possible. But in mathematical proofs it is quite common to "choose a fixed element" from a non-empty set. So how exactly do we achieve such a thing? Am I missing something?  Thank you in advance. I'm making a map where I want players to be able to choose there kits/classes. The problem I'm having with it though is giving players already enchanted Tools/Armour/Weapons.  I'm not sure if you can use the /give [player] [item] [amount] because you'll have to use the whole {} thing and I'm not sure how to make enchanted Tools/Armour/Weapons using that.  Can anyone help? Where to put the periods when using a parenthetical sentence? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? I'm currently reading Marcus' "Number Fields," and I'm having difficulty proving the following result:  Corollary 2.2: Let $m$ be a squarefree integer. The set of algebraic integers in the quadratic field $\mathbb{Q}(\sqrt{m})$ is $$ \left\{ a + b \sqrt{m} : a,b \in \mathbb{Z} \right\} $$ if $m \equiv 2,3 \pmod{4}$; and$$ \left\{ \frac{a+b\sqrt{m}}{2} : a,b \in \mathbb{Z} ~,~ a \equiv b \pmod{2} \right\}$$ when $m \equiv 1 \pmod{4}$. $\square$  It is easy enough to see that the claimed sets do, indeed, consist of algebraic integers. The other half of the proof proceeds something like this: suppose that $\alpha = r + s \sqrt{m}$ is an element of $\mathbb{Q}(\sqrt{m})$. Then $\alpha$ is a root of the polynomial $$ x^2 - 2rx + r^2 - s^2m $$ In particular, this means that $\alpha$ is an algebraic integer if and only if both $2r$ and $r^2 - s^2 m$ are integers. At this point, the proof stops, and says that this implies the result. And therein lies the problem: I don't see how to make the connection. I do know, for example, that if one of $r,s$ is an integer, then the other one must be, too, i.e., either both are integers, or both are not. However, I cannot get any further than that. Can anyone give me a gentle nudge in the right direction? Thank you very much! Touchpad stopped working 20.04 This is in very common use on the internet. I just read this:     "[The] hotel employee walks in, I say "uh, puppy" and she just NOPEd the [heck] out of the room."   This is often done by adding 'd to the word. How to install KDE along Unity? I have just installed the latest Ubuntu 14.04 and I would like to install KDE along side my Gnome/Unity that I have now. How can I do this?  Thanks! How To Show Node Comment Count in node.tpl? I have had the bright idea of changing my user permissions so as guests cannot see the comments on a node. I have no problems with Views around my site, as they can clearly see there are comments on nodes, and they will be more inclined to register. The problem seems to be when they arrive on my site directly to a node, they don't see any comment, and think "that's a bit crap; let's go."  Is there a way I could show the amount of comments below a node? What is the default password of the sudo? I visit some forums, it tells that the default password is password, but it doesn't work, how can I determine the password? Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$ Show that the function $g(x) = x^2 \sin\left(\frac{1}{x}\right) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$. I remember a story I have read buy could not remember the author the name or anything apart from the basic plot. The story is about some future where humanity has to fight with some aliens (not 100% sure). The story tells about a guy who was some kind of activist and who was in a prison two or more times and somehow helped humanity by being held in a prison (can't remember details). I think the story starts by saying that there is a statue in his honor that is not very well known but he did more to help humanity than famous politicians.   The story is most likely written by some famous English-speaking author, though I read it in a translation. I have a bunch of raw data values that are dollar amounts and I want to find a confidence interval for a percentile of that data.  Is there a formula for such a confidence interval? I was reading about the equipartition theorem and I got the following quotations from my books:  A diatomic molecule like oxygen can rotate about two different axes. But rotation about the axis down the length of the molecule doesn't count. - Daniel V. Schröder's Thermal Physics. A diatomic molecule can rotate like a top only about axes perpendicular to the line connecting the atoms but not about that line itself. - Resnick, Halliday, Walker s' Fundamentals of Physics.  Why is it so? Doesn't the rotation take place that way? Is it possible to tell a Valyrian steel sword from a very well crafted steel sword, by sight alone I'm sure in Westeros there are some incredibly well crafted steel swords, but could someone instantly spot a Valyrian steel sword over one of these?  Does Valyrian steel have any visual indications that it is Valyrian steel and not just a very well crafted steel sword?  I'm wondering, in canon, has anyone seen a sword and instantly recognised it as Valyrian steel.  For the purpose of this question, no one is allowed pick up or use the sword, so they cant feel it's weight, balance, sharpness, etc... Why is "cupboard" pronounced with a silent "p"? Why does as.factor return a character when used inside apply? KKT points and nonlinear optimality (theory) I have some questions about Karush Kuhn Tucker conditions. I am not clear about the theory. There are plenty of theorems and I am confused with it. Let us say we have the following problem: We have a general problem, not a convex problem. $$\min f(x)\\g_{j}(x)\le 0,\forall j\in J\\h_{k}(x)=0,\forall k \in K. $$ I will solve it.....Lagrange function, verifying primal feasibility, dual feasibility and complementary slackness and I will get some KKT points. Now what is the conclusion?  If I get no KKT point, can I say that the problem has no solution or what is the conclusion? If I get only one KKT point, is it a local minimum or even a global minimum? If I get more KKT points, what is the conclusion? In which case I will verify SOSC condition? If I get more KKT points or even for one point is it necessary? And the conclusion is what? The KKT points which satisfies the SOSC condition are strick local minimum and the poins which does not satisfy it are not the solution of the problem?  I am so sorry for so many questions, but any help would be very appreciated. Thank you very much in advance. Arrays, linked lists and other data structures in cmd.exe (batch) script Why on earth are o-scopes earth referenced? Thinking about it: You would never find a "Grounded" multimeter as robust and useful if a path to ground through the multimeter were introduced, modifying the circuit's behaviour and possibly damaging the multimeter with currents.  Why are so many oscilloscopes earth referenced? Upon reading some educational material, a majority of the "common mistakes made by students" are placing the grounding clip incorrectly and causing poor results - when the o-scope is just being used as a fancy voltmeter!  I've heard of a Tek scope having an isolation transformer within.. however ignoring that, and taking in to account that newer DSOs may have plastic cases (isolated from you most importantly I would assume) could I just remove the earthing pin, and install a 1:1 AC transformer inbetween the o-scope and outlet and be on my merry way probing various hot/neutral/earthed sources with no worries about a path to ground any longer through it?