$\mathbb{k}$ is a field. $\alpha \in \mathbb{k}$. Prove that if    polynomial $f (x)=x^{n} - 1$ has all the roots over $\mathbb{k}$, then   $g(x) = x^{n} - \alpha$ is either irreducible over $\mathbb{k}$ or    $\exists d&gt;1, d|n$, that $x^{d} - \alpha$ has at least one root over    $\mathbb{k}$   So there exist $n$ elements of $\mathbb{k}$ of order $n$. And they are closed under multiplication $(ab)^{n} = a^{n}b^{n} = 1$. We have $1^{n} = 1$. And $(a^{-1})^{n} = (a^{n})^{-1} = 1$. So it's a group under multiplication. I don't know if it helps in solving this problem, but that's all that i've figured out. What i should do with $x^{n} - \alpha$?
Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show that $x^n -a$ is irreducible over $F$?
how to install the DDE to the ubuntu 20.04? I have installed the Ubuntu 20.04, and want now I want to install Deepin Desktop Environment (dde)  My steps are:   add the ppa and install using apt  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update sudo apt install dde    But during the command sudo add-apt-repository ppa:leaeasy/dde I get the following error:  $ sudo add-apt-repository ppa:leaeasy/dde  You can install deepin desktop environment by run  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update apt-get install dde ------ Ubuntu Bionic: https://drive.google.com/file/d/1xkQPb4tWocknuYeY-t41W--kz-9SSL0V/view?usp=sharing ------ Important Issue: Q: Can dde on Ubuntu 16.04? A: Sorry, the dde depends on qt version above than 5.6.      If you want to install dde, please upgrade to 17.04 or above.  More info: https://launchpad.net/~leaeasy/+archive/ubuntu/dde Press [ENTER] to continue or Ctrl-c to cancel adding it.  Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                      Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                    Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease                                      Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease            Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release             404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   I get the similar error with sudo apt-get update :  $ sudo apt-get update Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                       Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                     Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease            Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release   404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   How can I install dde ? Is every metric space a subspace of some connected metric space? Is every metric space a subspace of some connected metric space? If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected.  I'm not sure whether this will work or not, but intuitively I think the answer is yes. Cannot modify files installed by `snap` I have installed a package using snap and I need to modify one of the files but when I try to change its ownership or permissions, I always get the following message:  sudo chmod +x ./my_file.js chmod: changing permissions of '/snap/my_app/my_file.js': Read-only file system   How can I modify files that are installed via snap? Is abstract algebra (mostly?) restricted to $2$-ary operators? Computational complexity of Fibonacci Sequence I understand Big-O notation, but I don't know how to calculate it for many functions. In particular, I've been trying to figure out the computational complexity of the naive version of the Fibonacci sequence:  int Fibonacci(int n) {     if (n &lt;= 1)         return n;     else         return Fibonacci(n - 1) + Fibonacci(n - 2); }   What is the computational complexity of the Fibonacci sequence and how is it calculated? My task is to write a precise mathematical statement that "the sequence $(a_n)$ does not converge to a number $\mathscr l$"  So, I have my definition of a convergent sequence: "$\forall\varepsilon&gt;0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|&lt;\varepsilon$ $\forall n \in \Bbb N$ with $n&gt;N$"  Would the correct negation of this be "$\forall\varepsilon&gt;0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|&gt;\varepsilon$ $\forall n \in \Bbb N$ with $n&gt;N$"?  It doesn't seem that this is the answer as the next part of my task is to prove that a sequence is divergent using my formed proof, but it'd be difficult to do since it's a general proof of divergence and not just a proof that $(a_n)$ doesn't converge a specific number $\mathscr l$  Perhaps I should find a prove that $(a_n)$ tends to $\pm\infty$? This is more simple but it does not include monotone sequences such as $x_n:=(-1)^n$.  Can someone assist me with this task? All comments and answers are appreciated. Find a bijection to prove a fact about integer partitions Prove that for fixed positive integers $k$ and $n$, the number of partitions of $n$ is equal to the number of partitions of $2n + k$ into $n + k$ parts.  I tried this with different values of n but can't seem to get a bijection between the two partitions. A matrix $M$ that commutes with any matrix is of the form $M=\alpha I$ Regression Proof that the point of averages (x,y) lies on the estimated regression line How do you show that the point of averages (x,y) lies on the estimated regression line? How to end paragraph text in an inverted pyramid shape? In many old books, the final paragraphs of chapters end in an inverted pyramid shape, like this:    I think this is beautiful, so I would like to imitate this for my book. I've found the code for a triangle in this post   \documentclass{article} \usepackage{microtype,shapepar}  \def\triangleupshape{% {0}% {0}b{0}\\% {8.66}t{-5}{10}\\% {17.32}t{-10}{20}\\% {17.32}e{0}% }      \begin{document} \shapepar{\triangleupshape}A ipsum dolor sit amet, consectetuer    adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. \par \end{document}   Which results in this:    But I'd like it to be inverted and the first part of the paragraph need to be full line width. Only the last few lines should end in an inverted pyramid like in the example above.  Is there a way to do this? Thanks in advance for your help. Which is the correct sentence?   The young cow is calf. The young cow is called calf. The young cow called as calf. I flashed new firmware and before doing so backed up apps and data with Titanium Backup. How do I restore? I have the folder on my PC which contains two .gz files and a properties file. Force MacBook to drain the battery while connected to charger I leave my MacBook Pro on my desk nearly all the time, connected to the charger with the lid closed, connected to an external display.  I've heard this is bad for the battery health, and everyone in my office have to replace swollen batteries every couple of years. I want to regularly discharge the battery, per , but I don't want to physically unplug it because that temporarily turns off the external monitor, and then I have to reconnect the monitor.  So I am looking for a software method of disabling the power input, to force the laptop to discharge its battery to 20% and then re-enable the power input. I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius.  I have managed to solve find the constants for $||.||_{1}$ and  $||.||_{2}$ but I cannot see how to continue if I want to prove the following:  $||A||_{\infty} \leq \sqrt{n}||A||_{2}$   and the ones for Frobenius:  $||A||_{F} \leq \sqrt{n}||A||_{1}$  $||A||_{F} \leq \sqrt{n} ||A||_{2}$  Any help is appreciated. Thank you for your time How to recover deleted files? How do I migrate my Keychain's "Local Items" to a new Mac? Why do we need multiple approvers for suggested edits? I already have the "Edit Questions And Answers" privilege. Yet, if I approve someone else's edit, I'm told that more votes will be required before the edit is accepted.  This seems a bit peculiar to me, since I could just copy the edited version and paste it myself. Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question Show that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.      Let $\phi$ be the homomorphism from $\Bbb R[x]$ onto $\Bbb C$ given by   $f(x) \rightarrow f(i)$ (that is, evaluate a polynomial in $\Bbb R[x]$   at $i$).  Then $x^2 + 1 \in \operatorname{Ker} \phi$ and is clearly a   polynomial of minimum degree in $\operatorname{Ker} \phi$.  Thus,   $\operatorname{Ker} \phi = \langle x^2 + 1 \rangle$ and $\Bbb R[x] /\langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.   My question is:  How is $x^2 + 1$ a polynomial of minimum degree in $\operatorname{Ker} \phi$? How to edit Parameters for an Object after it is created? When you first add a cylinder, you have the option to change the number of vertices, etc, but once you do anything else in blender, it seems you no longer can edit those parameters.   Is there a setting anywhere or an add-on that allows these parameters to be edited later on? If not, how easy would such an addon/feature be to create for anyone reading this? Too many of your edits were rejected, try again in 7 days