Let  $ \mathbb{Z}^{'}_{\mathbb{C}}$ be the integral closure of $ \mathbb{Z} $ in $ \mathbb{C} $, i.e. $ \mathbb{Z}^{'}_{\mathbb{C}}$ contains all complex roots of polynomials with integer coefficients and $1$ as the leading coefficient.  Prove that $ \mathbb{Z}^{'}_{\mathbb{C}} $ is not finitely generated as a $ \mathbb{Z} $-module.  My attempt: Suppose $ \mathbb{Z}^{'}_{\mathbb{C}}=\mathbb{Z}\alpha_{1}+...+\mathbb{Z}\alpha_{n} $ for some $ \{ \alpha_{i}\}_{i=\overline{1,n}} \subset \mathbb{Z}^{'}_{\mathbb{C}} $.  Let's take $ \alpha , \beta \in \mathbb{Z}^{'}_{\mathbb{C}} $ with minimal polynomials of degrees $ m $ and $ n $ respectively. We can construct a monic polynomial with integral coefficients $ P(z)=\prod(z-(\alpha_{i}+\beta_{j})) $ where $ \alpha_{i},\beta_{j} $ are the conjugates of $\alpha $ , $\beta $ and therefore the minimal polynomial of $ \alpha + \beta $ has degree $ \leq mn $.   I was thinking that the degrees of the minimal polynomials of the elements of $ \mathbb{Z}\alpha_{1}+...+\mathbb{Z}\alpha_{n} $ could be somehow bounded by a certain constant, whereas $ \mathbb{Z}^{'}_{\mathbb{C}}$ contains elements whose degrees of the minimal polynomials are arbitrarily large. I think this works when we are asked to prove that the set of all algebraic numbers $ \overline{\mathbb{Q}} $ is not a finitely dimensional $\mathbb{Q}$-vector space, but I don't know how to proceed in my case.  I would really appreciate any ideas, hints or solutions. Thank you very much!
Let     $$ \mathbb{Z}^{'}_{\mathbb{C}}=\{ z \in \mathbb{C} | \exists f \in \mathbb{Z}[X] \text{ monic such that } f(z)=0\} $$   be the integral closure of $ \mathbb{Z} $ in $ \mathbb{C} $. Prove that $ \mathbb{Z}^{'}_{\mathbb{C}} $ is not finitely generated as a $ \mathbb{Z} $-module.  My attempt: Suppose $ \mathbb{Z}^{'}_{\mathbb{C}}=\mathbb{Z}\alpha_{1}+...+\mathbb{Z}\alpha_{n} $ for some $ \{ \alpha_{i}\}_{i=\overline{1,n}} \subset \mathbb{Z}^{'}_{\mathbb{C}} $. I know that given $ \alpha , \beta \in \mathbb{Z}^{'}_{\mathbb{C}} $ with minimal polynomials of degrees $ m $ and $ n $ respectively, then we can construct a monic polynomial with integral coefficients $ P(z)=\prod(z-(\alpha_{i}+\beta_{j})) $ where $ \alpha_{i},\beta_{j} $ are the conjugates of $\alpha $ , $\beta $ and therefore the minimal polynomial of $ \alpha + \beta $ has degree $ \leq mn $.   I was thinking that then the degrees of the minimal polynomials of the elements of $ \mathbb{Z}\alpha_{1}+...\mathbb{Z}\alpha_{n} $ could be somehow bounded by a certain constant, whereas $ \mathbb{Z}^{'}_{\mathbb{C}}$ contains elements whose degrees of the minimal polynomials are arbitrarily large, take $ 2^{\frac{1}{n}} $ for example,$ n\geq 2$. I think this works when we are asked to prove that the set of all algebraic numbers $ \overline{\mathbb{Q}} $ is not a finitely dimensional $\mathbb{Q}$-vector space, but I don't know how to proceed in my case.  I would really appreciate any ideas, hints or solutions. Thank you very much!
How to change password using usermod? I am superadmin of a server and like to change password of an existing user . How can I do that ?  I tried   usermod -p 'new-password' john   but it didn't worked ? How can I change font rendering to sub-pixel BGR ordering? FM Transmitter oscillation problem I've been trying to get this simple FM transmitter to work over the past few days and I could use some help in a few areas.      First iteration was on a breadboard with all parts included.  The exception being I didn't have a 0.1uH coil handy but I had another coil rated at 0.7uH (using my LCR meter).  I also didn't have a C4 value of 4.7pF but I had a 10pF value.  I figured I could just tune it later.  I hooked a scope up to it and got no oscillation.  I tried connecting the probe to the gate and the collector separately and together.  Still no oscillation.  Second iteration.  I removed the audio component of the circuit.  Essentially everything to the left of C2.  I had gotten that working no problem anyway so I figured I would concentrate on the oscillator.  This time I tried different values for the inductor ranging from 20uH to 1mH and different fixed values for the variable cap (10pf, 20pf, 30pf).  These were all ceramic caps. No love.  Third iteration. I tested the transistor to make sure that wasn't the issue.  Seems fine.  PN2222A variety. (Also tried 2N3904 and BC547). I then modeled this up on a piece of perf board.  I even found a SMD inductor of 0.1uH so now I thought I would be good.  Tested all the pieces on the meter before installation. Still nothing. I'm not seeing anything on the spectrum analyzer either.  Questions:   Do you think my benchtop power supply is causing a problem?  All the various versions of this type of FM transmitter circuit on YouTube and the Internet all seem to use a battery.  Any chance the 60Hz main power is messing with things?  I've put .1uF and higher caps on the power rails. I have a 200Mhz SIGLENT SDS1202X-E scope.  I've built and tested other oscillators up to a few MHz before with no issues.  Any thoughts on that being the issue?  I would expect to see something on the scope right? Any chance the probe is dampening the oscillations?     I know there are a lot better FM transmitter circuits out there I was just experimenting with this (and others like it) and couldn't get any to work. My Prof wrote that   $\sum_{n=2}^{\infty} n(n-1)x^{(n-2)} = \frac{2}{(1-x)^3}$  when $-1&lt; x&lt;1$  I do not understand where the above formula comes from. Can someone please explain the steps or provide a hint?   Thanks! Dynamic loading of python modules I used a website called Seeking Arrangements and met a  sugar daddy. He asked for my email login, online banking login, routing, account number, Facebook, as well as my  address which is the only thing I gave him since it’s a PO. I’ve been extremely vague just to see if this is real.   I don’t plan to give him anything because I have only been talking to him for a few months and  I feel conflicted. I doubt that he is trying to scam me but I suspect that he has some kind of master plan. With things such as Venmo, Zelle, and Cashapp, what is the need for “put me on his payroll?”   He claims that he is in another country on a business trip.  He said that  he tried to send me a gift but he lost access to his card. He is now asking to deposit more than 10k into my account and have me wire it. Is this a l classic way of scamming people? If so, is it common that they spend months creating a relationship with you  to catfish you? Different dependencies for different build profiles I attended a business meeting in the Midwest and one of our clients took our management team and a few other business partners to dinner at an awesome Mexican restaurant. When the appetizers were served I commented that it was a wonderful presentation and that everything looked very fresh, including the avocado slices.   Speaking to the woman seated across from me I said that even though I use a good amount of lemon juice, whenever I prepare avocados they still turned dark pretty quickly. This happens whether they are sliced, used in guacamole, etc. She responded that the secret is to use lime juice. She proceeded to tell me that she was born and raised in Central America and that it is customary there to use fresh limes or juice rather than lemon.  I have tried it and lime seems to work much better than lemon, so it would stand to reason that they are more acidic. However, I never thought there was that much difference. So, on to the questions.  Are limes more acidic than lemons? Are there any other differences in their properties that could make a difference? A closed subset of an algebraic group which contains $e$ and is closed under taking products is a subgroup of $G$ A closed subset of an algebraic group which contains $e$ and is closed under taking products is a subgroup of $G$.   Denote this set as $X$. If the condition of $X$ being closed is dropped, this statement does not hold. The set of nonzero integers in $\mathbb{G}_m$ over $\mathbb{C}$ is a counterexample.  It suffice to prove that for any $x \in X$, $x^{-1}X = X$. As $X$ is closed under taking products, it is clear that $X \subseteq x^{-1}X$. In order to prove the inverse inclusion, the closedness of $X$ (under Zariski topology) must be used.   Let $\phi: G \rightarrow G, y \mapsto x^{-1}y$ is a homoemorphism of $G$ as an algebraic variety. So $x^{-1}X = \phi(X)$ is a closed subset of $G$ containing $X$. But Why are they equal?  Thanks very much. Is there a list of error codes for Minecraft? Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved. $C(n,p)$: even or odd? How do I make an Bootable Ubuntu USB containing the the latest version of Ubuntu?  For completeness, how would I be able to do this on Ubuntu, Windows, OS X or another Linux distro? Why can a function modify some arguments as perceived by the caller, but not others? I'm trying to understand Python's approach to variable scope. In this example, why is f() able to alter the value of x, as perceived within main(), but not the value of n?  def f(n, x):     n = 2     x.append(4)     print('In f():', n, x)  def main():     n = 1     x = [0,1,2,3]     print('Before:', n, x)     f(n, x)     print('After: ', n, x)  main()   Output:  Before: 1 [0, 1, 2, 3] In f(): 2 [0, 1, 2, 3, 4] After:  1 [0, 1, 2, 3, 4] While using the Pixel Inspector, whenever I click "copy all values" it only copies values that you can see on the screen. Anyone know a solution? I need all the values to count the number of times 1, 2, and 0 appear on the graph. How to append something to an array? How do I append an object (such as a string or number) to an array in JavaScript? In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational,  and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but I am curious as to whether anyone has a method for proving this without making use of the Taylor Series Expansion. I feel as though doing so must be possible using some number theory, but am low on ideas as to an alternative approach to the question.  Note:  If anyone is interested in my solution using Taylor Series Expansion (although it is not the focus of my question), here it is :      From    $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} -  \dots$$   We see that    $$ \alpha = \sin 1 = 1 - \frac{1^3}{3!} + \frac{1^5}{5!} -  \dots $$   Given integers $a$ and $b$, if $\alpha = \frac{a}{b}$ then it follows that $b!\alpha \in \mathbb{Z}$, and $b!\alpha = C + D$ where $C \in \mathbb{Z}$ and we have :   $$ D = \begin{cases} \pm(\frac{1}{b+1} - \frac{1}{(b+1)(b+2)(b+3)} + \dots) \text{    if $b$ is even}\\ \pm(\frac{1}{(b+1)(b+2)} - \frac{1}{(b+1)\dots(b+4)} + \dots ) \text{ if $b$ is odd}\\ \end{cases} $$   In each case we can see that $0 &lt; D &lt; 1$, giving us a contradiction.    Thus, we have that $\sin 1 $ is irrational.    $$ \blacksquare $$ Per   I did the following:  ec2run ami-3c994355 --region us-east-1 -n 1 -t m1.large -z us-east-1d   On the machine:  sudo apt-get update sudo apt-get -y -o Dpkg::Options::="--force-confdef" -o Dpkg::Options::="--force-confold" dist-upgrade   I still get a prompt asking me which config file I want to use. These are the lines that come before the prompt:  Setting up grub-pc (1.99-21ubuntu3.1) ...   then:                           ┌───────────────────────────────────────────────────────┤ Configuring grub-pc ├───────────────────────────────────────────────────────┐                                                        │ A new version of configuration file /etc/default/grub is available, but the version installed currently has been locally modified.  │                                                        │                                                                                                                                     │                                                        │ What do you want to do about modified configuration file grub?                                                                      │                                                        │                                                                                                                                     │                                                        │                                     install the package maintainer's version                                                        │ I am trying to install Ubuntu 16.04 LTS on a Dell XPS 13 (9350). The laptop came with Windows 10 pre-installed. I am installing from a pendrive but I think the issue would be the same if I used e.g. a USB DVD drive.   Dude, where's my hard drive?  I am blocked because the installation program cannot find 8.6Gb of free space on my hard drive despite the fact that there are hundreds of Gb free according to Windows. Digging a bit into the problem, it seems that Ubuntu cannot see my hard disk at all.    This is the output of GParted:    I am not sure what that drive mounted at /cdrom is, but it seems the only device that exists in the system that is not a RAM disk or a loop device:   Content of /proc/partitions:   ubuntu@ubuntu:~$ cat /proc/partitions   major minor  #blocks  name     1        0      65536 ram0     1        1      65536 ram1     1        2      65536 ram2     1        3      65536 ram3     1        4      65536 ram4     1        5      65536 ram5     1        6      65536 ram6     1        7      65536 ram7     1        8      65536 ram8     1        9      65536 ram9     1       10      65536 ram10     1       11      65536 ram11     1       12      65536 ram12     1       13      65536 ram13     1       14      65536 ram14     1       15      65536 ram15     7        0     130048 loop0     7        1    1104188 loop1     8        0    1970239 sda     8        1    1953439 sda1   Summary of fdisk -l:   ubuntu@ubuntu:~$ sudo fdisk -l |grep Disk Disk /dev/ram0: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram1: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram2: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram3: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram4: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram5: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram6: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram7: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram8: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram9: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram10: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram11: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram12: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram13: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram14: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram15: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/loop0: 127 MiB, 133169152 bytes, 260096 sectors Disk /dev/loop1: 1.1 GiB, 1130688512 bytes, 2208376 sectors Disk /dev/sda: 1.9 GiB, 2017525248 bytes, 3940479 sectors Disklabel type: dos Disk identifier: 0x00000000   Result of fdisk of the only non-loop non-RAM device (not sure this is relevant):   ubuntu@ubuntu:~$ sudo fdisk -l |grep -A 50 sda Disk /dev/sda: 1.9 GiB, 2017525248 bytes, 3940479 sectors Units: sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disklabel type: dos Disk identifier: 0x00000000  Device     Boot Start     End Sectors  Size Id Type /dev/sda1  *      129 3907007 3906879  1.9G  c W95 FAT32 (LBA)   I am blocked at this point. Any help will be much appreciated.   Additional information (following comments):   Running dmesg |grep -i sata shows (among others) eight lines that suggest some kind of error:   ata1: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata2: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata3: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata4: SATA link down (SStatus 0 SControl 0) ata5: SATA link down (SStatus 0 SControl 0) ata6: SATA link down (SStatus 0 SControl 0) ata7: SATA link down (SStatus 0 SControl 0) ata8: SATA link down (SStatus 0 SControl 0)  I have "Restarted" windows instead of only "Shutdown" it to avoid problems with "fast startup", following some recommendations I have read of other users having problems installing Linux on Windows 10 laptops.ago   Regarding SATA, the BIOS says: "SATA Operation: RAID ON". The other two options available are "Disabled" and "AHCI".    I have followed the instructions of these two pages from Dell: , . How to initialize a struct in accordance with C programming language standards I want to initialize a struct element, split in declaration and initialization. This is what I have:  typedef struct MY_TYPE {   bool flag;   short int value;   double stuff; } MY_TYPE;  void function(void) {   MY_TYPE a;   ...   a = { true, 15, 0.123 } }   Is this the way to declare and initialize a local variable of MY_TYPE in accordance with C programming language standards (C89, C90, C99, C11, etc.)? Or is there anything better or at least working?  Update I ended up having a static initialization element where I set every subelement according to my needs. I have one bash source run.sh as follows,  #!/bin/bash if [ $# -ne 1 ]; then     exit fi ...   when I execute it in two ways, there are different behaviors. The first way is,  source run.sh   It will close the terminal after execution. The second way is,  ./run.sh   this will simply finish running the script, and stay on the terminal. I am asking if there is a command for exiting a bash scripts for both source run.sh and ./run.sh execution. I have tried return too, which does not work well under ./run.sh execution.  More generally, I am interested in why this is happening, and what's difference between using "source" and "." for script execution?