Problem  How many elements are there in $\mathbb{Z}[i]/I$ where $I= \langle2+2i\rangle$  $2+2i + I =I$ . So $2+2i$ is equivalent to $0$ . So $2+2i=0$ which implies $i=-1$ which inturn implies $2=0$.  How to move after that?
The question is :   Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic.  My try:  I started with elements $2, 1+i \in \mathbb Z[i]$. Clearly, the product is $0$ in the coset representation.  The part where i am struck at is the number of elements and the characteristic.  Well, $2+2i+\langle 2+2i\rangle=0+\langle 2+2i\rangle$  Can we directly write $i^2=-1$ in the coset representation?
Is two-thirds plural? Is 2/3 always, sometimes or never plural?  E.g.     1a) 2/3 of the pizza were eaten.     1b) 2/3 of the pizza was eaten.      2a) 2/3 of the visitors were men.     2b) 2/3 of the visitors was men.   I feel that example 1 could go either way but example 2b sounds very wrong.  Is there a rule for this? Calculating grid convergence (True North to Grid North) I would like to know how to calculate the grid convergence angle between True North and Grid North for UTM maps. I am using ArcGIS 10.0 (ArcView licence). and have used the  tool from the Cartography toolbox. However, I would like to know how to calculate this angle myself (using Excel) to compare against the ArcGIS result? Omitting "and" in a sentence He called her, emailed her, texted her, tweeted her—all to no use.   Strictly speaking, I would need to write texted her and tweeted her, but I'm omitting and to convey a rhythm and sense of urgency. What do you call this kind of construction, and is it accepted grammar (at least for creative writing, if not in an academic context)? Infinitely differentiable function with given zero set? How do I edit a file in /proc that can be viewed with cat, but is not editable by mousepad and abiword? I've noticed that Visual Studio 2008 is placing square brackets around column names in sql. Do the brackets offer any advantage? When I hand code T-SQL I've never bothered with them.  Example:  Visual Studio:  SELECT [column1], [column2] etc...   My own way:  SELECT column1, column2 etc... I'm usually running many different StackExchange sites in Firefox using tabs (using ReloadEvery to update every 5 minutes).  Right now to check for activity, I have to manually click on the tab to see if the envelope is lighted.  How about making the icon on the tab i.e. the title text change in some way, denoting that you have activity on the page?  We already add a message count for the chat tabs, so this shouldn't be very complicated.  Possible Example: Total order on complex numbers Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $. Cutting features to do opposite of clipping in ArcGIS Desktop? What´s the easiest/smartest way of getting the opposite output of the clipping tool?   I just need the area of feature A1 which is not covered by feature A2 in a new Layer/File.  Now, I get the result by union´ing both features and deleting the "wrong" parts by hand.  I'm using ArcGIS Desktop 10. How do you make the camera in blender cycles render everything so it looks like pixels? I want everything in my scene to look pixelly like this image I found: \phantom causes a spacing issue between paragraghs When did the word "so" begin to be used to start a sentence? In the last few years, I've noticed a growing usage of the word "so" to begin a sentence, especially in the context of higher education.  For example:   Interviewer: "What is the nature of your research"  Researcher: "So, what we wanted to find out is..."  It seems to be a replacement the word "well", or, more informally, "ok".  Has this usage of the word been around for a long time and I'm just now noticing it?  Do you think that is a valid use of the word? How to upgrade Git on Ubuntu Hardy? Multiple runnable classes inside JAR, how to run them? I am using UBUNTU 18.10 and I had problem with TL-WN822n v5 driver in UBUNTU and Kali (4.19.0-kali5-amd64)  The problem is driver=rtl8xxxu so I found a driver in this page :   and installed the driver on Kali and it worked perfectly fine but when I wanted to install it on my ubuntu, after I ran the sudo dkms install rtl8192eu/1.0; I got this error :   Kernel preparation unnecessary for this kernel.  Skipping...  Building module: cleaning build area... 'make' all KVER=4.18.0-25-generic...(bad exit status: 2) ERROR (dkms apport): binary package for rtl8192eu: 1.0 not found Error! Bad return status for module build on kernel: 4.18.0-25-generic (x86_64) Consult /var/lib/dkms/rtl8192eu/1.0/build/make.log for more information.   and This is my Log file (/var/lib/dkms/rtl8192eu/1.0/build/make.log) :  DKMS make.log for rtl8192eu-1.0 for kernel 4.18.0-25-generic (x86_64) Sun Jul  7 13:35:02 +0430 2019 make ARCH=arm64 CROSS_COMPILE= -C /lib/modules/4.18.0-25-generic/build  M=/var/lib/dkms/rtl8192eu/1.0/build  modules make[1]: Entering directory '/usr/src/linux-headers-4.18.0-25-generic'   CC [M]  /var/lib/dkms/rtl8192eu/1.0/build/core/rtw_cmd.o gcc: error: unrecognized command line option ‘-mlittle-endian’; did you mean ‘-fconvert=little-endian’? make[2]: *** [scripts/Makefile.build:325:/var/lib/dkms/rtl8192eu/1.0/build/core/rtw_cmd.o] Error 1 make[1]: *** [Makefile:1534:_module_/var/lib/dkms/rtl8192eu/1.0/build] Error 2 make[1]: Leaving directory '/usr/src/linux-headers-4.18.0-25-generic' make: *** [Makefile:1710: modules] Error 2   What's the problem ?? Hibernate JPA Sequence (non-Id) Is it possible to use a DB sequence for some column that is not the identifier/is not part of a composite identifier?   I'm using hibernate as jpa provider, and I have a table that has some columns that are generated values (using a sequence), although they are not part of the identifier.  What I want is to use a sequence to create a new value for an entity, where the column for the sequence is NOT (part of) the primary key:  @Entity @Table(name = "MyTable") public class MyEntity {      //...     @Id //... etc     public Long getId() {         return id;     }     //note NO @Id here! but this doesn't work...     @GeneratedValue(strategy = GenerationType.AUTO, generator = "myGen")     @SequenceGenerator(name = "myGen", sequenceName = "MY_SEQUENCE")     @Column(name = "SEQ_VAL", unique = false, nullable = false, insertable = true, updatable = true)     public Long getMySequencedValue(){       return myVal;     }  }   Then when I do this:  em.persist(new MyEntity());   the id will be generated, but the mySequenceVal property will be also generated by my JPA provider.  Just to make things clear: I want Hibernate to generate the value for the mySequencedValue property. I know Hibernate can handle database-generated values, but I don't want to use a trigger or any other thing other than Hibernate itself to generate the value for my property. If Hibernate can generate values for primary keys, why can't it generate for a simple property? How would one denote the set of all subsets of $A$ which have size $2$?  Would  $$\binom{A}{2}$$  be a good idea? Isomorphism between two models of Hilbert's Axioms with different notions of congruence I've been reading about Hilbert's Axioms of geometry, and I have two models on $\mathbb{R}^2$ which satisfy the usual axioms of incidence, betweenness, and congruence.   The first model has a distance function given by  $$ d(A,B)=|a_1-b_1|+|a_2-b_2| $$ where $A=(a_1,a_2)$, $B=(b_1,b_2)$, and two segments $AB\cong CD$ if $d(A,B)=d(C,D)$.  The second model has a distance function given by $$ d(A,B)=\sup\{|a_1-b_1|,|a_2-b_2|\}, $$ and again $AB\cong CD$ if $d(A,B)=d(C,D)$.  The author claims that these two models are isomorphic in that there is a bijection between the sets of points in $\mathbb{R}^2$ that preserves lines, betweenness and congruence. I've been toying with it for a while, but it seems to me that the notions of congruence in particular are too different.   For example, in the first model, if $A=(0,0)$ and $B=(2,1)$, and $C=(2,0)$ and $D=(5/2,5/2)$, then $AB\cong CD$ since both segments have length $3$. To preserve congruence, I suppose two segments are congruent if the maximum of the displacement along the $x$-axis or $y$-axis by traveling along one segment is equal to the maximum of the displacement along either axis when traveling along the other segment. In this sense, how could you shape the segment $CD$ so that the maximum of the displacements along either axis is the same as the maximum of the displacement for whatever the image of $AB$ is?  In the first model, $AB$ is congruent to any segment which is the hypotenuse of triangle with legs that sum to $3$, and this sum could split into any two terms. I just don't see a function that could handle all these possibilities at once. Does anyone know of a particular isomorphism between these two? Thanks. The $n$ complex $n$th roots of a complex number $z$ Suppose $z$ is a nonzero complex number, so $z=re^{i\theta}$. Show that there are only $n$ distinct complex $n$-th roots, given by $r^{1/n}e^{i(2\pi k+\theta)/n}$ for $0\leq k\leq n-1$.   My proof: Let $z=re^{i\theta}\in\mathbb{C}$ and $a\in\mathbb{C}$ s.t. $a^{n}=z$. Let $a=\rho e^{i\varphi}$. Then, $a^{n}=(\rho e^{i\varphi})^{n}$. Setting $\rho^{n}e^{i\varphi n}=re^{i\theta}$ and rewriting in polar form, we get $$\rho^{n}(\cos(n\varphi)+i\sin(n\varphi))=r(\cos(θ)+i\sin(\theta)).$$ Thus $\rho^{n}=r$ $\rightarrow$ $\rho=r^{1/n}$ and $n\varphi=\theta\rightarrow\varphi=\theta/n$. Thus the roots of $a\in\mathbb{C}$ have the form $$r^{1/n}(\cos(\tfrac{\theta}{n})+i\sin(\tfrac{\theta}{n}))=r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))$$ for some $j\in\mathbb{Z}$ due to the periodicity of trig functions. Then, $$r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))=r^{1/n}e^{i\left(2\pi j+\tfrac{\theta}{n}\right)}=r^{1/n}e^{i(\theta+2\pi k)/n}$$ for $0\leq k\leq n-1 ,j=nk\in\mathbb{Z} $.  I feel like there could be a more concise way to put this.