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Irreducibility of a polynomial if it has no root (Capelli) Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$.   This  to a related question mentions the result is due to Capelli.  I can prove the result if $F$ has characteristic $p$ as follows.  Suppose $f$ is reducible: $f(X)=g(X)h(X)$ with $g(X)$ an irreducible factor of degree $m$, $1\le m&lt;p$. Then if $\alpha$ is a root of $g$ in some extension field $K$ of $F$, we have $$f(X)=X^p-\alpha^p=(X-\alpha)^p$$ so its divisor $g(X)$ must be of the form $(X-\alpha)^m$. Since the coefficient of $X^{m-1}$ in $g$ is in $F$, we have $m\alpha\in F$.  So $\alpha\in F$ because $m$ is invertible modulo $p$.  How would you show the result in other characteristics? This is the proof, which I mostly understand except for one bit:  You have $h_1 \in H_1$ and $h_2 \in H_2$.   We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in h_2^{-1}H_1h_2 = H_1$. Similarly, we have $(h_1^{-1}h_2^{-1}h_1)h_2 \in H_2$. Therefore  $$ h_1^{-1}h_2^{-1}h_1h_2 \in H_1 \cap H_2 = \{1_G\} $$  and so $h_1^{-1}h_2^{-1}h_1h_2 = \{1_G\}$. Let's first multiply everything on the left by $h_1$  $$h_1^{-1}h_2^{-1}h_1h_2 = \{1_G\}$$ $$ h_1 h_1^{-1}h_2^{-1}h_1h_2 = h_1 \{1_G\}$$ $$ h_2^{-1}h_1h_2 = h_1 \{1_G\}$$   Multiply both sides on the left by $h_2$ giving us  $$ h_1h_2 = h_2 h_1 $$  The bit I don't get is right at the beginning. Why is this correct: "We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in h_2^{-1}H_1h_2 = H_1$." Cannot use polyglossia/babel with biblatex in LuaTeX: etoolbox Toggle undefined This is an example test.tex document. Both babel and polyglossia failed and produce the same error: \documentclass{book}  %\usepackage[vietnamese]{babel} \usepackage{polyglossia} \setdefaultlanguage{vietnamese}  \usepackage{biblatex}   \title{Sth}  \begin{document}  \maketitle  \end{document}   is the full log. And my config for VS Code: &quot;latex-workshop.latex.recipes&quot;: [     {         &quot;name&quot;: &quot;lualatex-&gt;biber&quot;,         &quot;tools&quot;: [             &quot;lualatex&quot;,             // &quot;biber&quot;,         ]     } ], &quot;latex-workshop.latex.tools&quot;: [     {         &quot;name&quot;: &quot;lualatex&quot;,         &quot;command&quot;: &quot;lualatex&quot;,         &quot;args&quot;: [             &quot;--shell-escape&quot;,             &quot;-synctex=1&quot;,             &quot;-interaction=nonstopmode&quot;,             &quot;-file-line-error&quot;,             &quot;-pdf&quot;,             &quot;%DOC%&quot;         ]     },     {         &quot;name&quot;: &quot;biber&quot;,         &quot;command&quot;: &quot;biber&quot;,         &quot;args&quot;: [             &quot;%DOCFILE%&quot;         ]     } ],  I installed all Vietnamese-related package I found using tlmgr search, notably babel-vietnamese and vntex, to no avail. When I use English language, it works. When I paste the exact document to Overleaf, it also works. So it's something on my part. How can I fix this? use contradiction to prove that the square root of $p$ is irrational sum of the residue at all poles of the rational function p/q is 0. We are given two polynomials p, q with deg(q)>1+deg(p). Show that the sum of the residue at all poles of the rational function p/q is 0.   How do I go about solving this? Forbid links to open new tabs in Google Chrome OS: Windows 7. I want install Linux Ubuntu on portable external Hard Drive USB 2.0. I read a variety of methods, but I'm confused by contradictory advices and methods. Actually, installing Ubuntu on anything other than clean PC looks much more complicated than installing Windows. I need to have Ubuntu package + Python 2.7 + C compiler installed. My external hard drive is partially filled, I need to keep all files.   Will Ubuntu run entirely on external hardrive, without writing anythinhg to Windows system folders and to registry?  Can anyone post detailed guide how install Ubuntu on external hard drive? Find the maximum value of $ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$ How to compress / reduce the size of JPEG photos for archiving? I have 50,000 high resolution JPEG photos, where a couple of them might occasionally be needed, about once a year.  I wanted to zip them to save disk space, except that zipping gives no space benefit - so trying to reduce the images disk usage using winzip, winrar or 7zip was not successful.  Is there any software or algorithm similar to zip to compress image size on hard disk for storage without loosing any image information? I have an issue which I've been struggling with today. The JSON is returned back to me with a project Id as the key, this Project Id is dynamically generated and due to this I have not been able to come up with a way to deserialize this response. Any suggestions?   {      "success":true,    "numRows":5,    "data":{         "23000079":{            "projectId":"23000079",          "name":"Mcdonalds",          "departmentId":"9383",          "responsibleUserId":"7444434"       }    } } Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms? Context: Rings.     Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?   Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck... I'm going to install 64 bit 14.04 on a new machine with something like 1 to maybe 128GB RAM and even more RAM later. How much space should I allocate to the swap partition?   And especially: WHY those recommendations for these numbers?    (None of the existing answers  and  explain any of the why and are a bit dated if we look at 1GB or RAM and up) Let $m\in \mathbb{N}$ and $c_1,c_2,\ldots,c_m \in \mathbb{R}_+$. Show that $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$$  My attempt: Since $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} \leq \lim_{n\rightarrow \infty}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}} = \lim_{n\rightarrow \infty} \sqrt[n]{n}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}}=\lim_{n \rightarrow \infty} \max\{\sqrt[n]{c_1^n},\sqrt[n]{c_2^n},\ldots,\sqrt[n]{c_m^n}\}=\lim_{n \rightarrow \infty}\max\{c_1,c_2,\ldots,c_m\}=\max\{c_1,c_2,\ldots,c_m\}$$  it follows that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n}$ is bounded, but I don't think it's monotonically decreasing, at least I can't prove this. Can anybody tell me whether the approach I have chosen is a good one, whether what I have done is correct and how to finish the proof? How do I use an NBT editor to modify item properties? How can I make a time delay in Python? I would like to know how to put a time delay in a Python script. How do  I create a desktop wallpaper slideshow with my own set of images?   With the images coming from a pictures folder, or pre-selected With the images displayed in order, or randomly With the images changing every few minutes, or after random periods of time How do I enable the "partner" repository? I keep seeing places refer to the "partner" repository as a place I can get software, how can I enable this repository? Please specify how to do this graphically and via command line. How to install Anaconda on Ubuntu? How to install  on Ubuntu?   Is there a way to use apt-get install?   I only have command line access to my server. How do I install Anaconda on Ubuntu 14.04 from the command line? So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. However, I do not know what $\lfloor i\rfloor$ equals. Is there a clearly defined definition of $\lfloor i\rfloor$? How to both display a command line's output on console and save the output into a text file?