How to solve $x^5\equiv 3 \pmod{19}$?
Solve $x^{5} \equiv 2$ mod $221\ $ [Taking modular $k$'th roots if unique]
If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field?  I am trying to prove that the ring of fraction $L(\alpha)$ is equal to $L[\alpha]$. To do this, I use an exercise done in class : Let $K$ a field, $\alpha \in K$ and $L$ a subfield of $K$. Then $L(\alpha)$ is the smallest subfield of $K$ containing $L$ and $\alpha$.  Is anyone could help me? Pokemon Go: Can't log in? Are everyone unable to log in right now? 2016 July 16. Where I live none can log in to the servers at the moment, and it all happened after the sudden release of the game in my country.   As I understand the game was recently released in 26 countries, so I guess they have problems and had the servers go offline in order to fix the issues. How do I tokenize a string in C++? Java has a convenient split method:  String str = "The quick brown fox"; String[] results = str.split(" ");   Is there an easy way to do this in C++? I have been trying to add some boarding passes to wallet. I have downloaded the airline app and logged in. I have tried to "scan the code" as suggested in wallet, but the only code is a QR code and nothing happens when I try to scan. This should be that difficult but I cannot figure it out. Is there an easy way to get this done? Please explain to me how to do it. Thanks. Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$. Let $S$ be a bounded set. Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$. Note: If $\sup S$ is in $S$, it’s sufficient to define $s_n = \sup S$ for all $n$.  What I have tried so far:  Let $M=\sup S, L = \lim\, s_n,\,|s_n-L|&lt;\epsilon/2$, then $M-\epsilon &lt; L- \epsilon/2 &lt; s_n &lt; M &lt; M+\epsilon$. But I get this based on $M &lt;= \epsilon+L$, because $\epsilon+L$ is an upper bound for $s_n$.   However, as $s_n$ are just points in $S$, so I don't know if it's sufficient to say $M$, which is the $\sup$ of $S$, is smaller than $L+\epsilon$. Create an item in sharepoint calender when infopath form is submitted in sharepoint 2010 through workflow How to control the coordinates in Tikz & Matrix? I use Tikz with Matrix because I need to draw arrows and write on and beside the matrix. However, I don´t control the coordinates well other than using trail and error. For instance in the matrix below I have put an X at (0,0) and it is not in the centre of the matrix. However, if the matrix is 4x4 than it is in the centre. Is there any convenient way to know the coordinates? \documentclass[border=1cm]{standalone}  \usepackage{tikz} \usetikzlibrary{matrix}   \begin{document}  \begin{tikzpicture} \matrix(m) [matrix of math nodes, nodes in empty cells, nodes={minimum size=1cm, outer sep=0pt, text height=1.5ex, text depth=.25ex}] {  36     &amp;   &amp; 32    &amp;   &amp; 33    &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  10     &amp;   &amp; 8     &amp;   &amp; 7     &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  12     &amp;   &amp; 17    &amp;   &amp; 16    &amp;   \\         &amp;   &amp;       &amp;   &amp;       &amp;   \\  23     &amp;   &amp; 15    &amp;   &amp; 16    &amp;   \\     &amp;   &amp;   &amp;       &amp;   &amp;       &amp;   \\ };  \draw (m-1-1.north west) rectangle (m-8-6.south east);    % Lineas horizontales \foreach \i in {2,4,6,8} {   \draw (m-\i-1.south west) -- (m-\i-6.south east); }  % Lineas verticales \foreach \j in {2,4,6} {   \draw (m-1-\j.north east) -- (m-8-\j.south east); }  % dual ui \node [font=\color{red}] at (3.5,3) {$\ge 10$}; \node [font=\color{red}] at (3.5,1) {$\ge 8$}; \node [font=\color{red}] at (3.5,-1) {$=4$}; \node [font=\color{red}] at (3.5,-3) {$=3$};  % dual vj \node [font=\color{red}] at (-2.5,-5) {$\le12$}; \node [font=\color{red}] at (-0.5,-5) {$=12$}; \node [font=\color{red}] at (1.5,-5) {$=6$};  \node [font=\color{red}] at (0,0) {X};  \end{tikzpicture}     \end{document} I accidentally rsynced .thunderbird to user@host instead of to user@host:/home/user.  Where is .thunderbird now?  Is there a good way to search for the folder? What if an abbreviation is at the end of a declarative sentence? Should I duplicate a full stop sign? What if an abbreviation is at the end of a declarative sentence? Should I duplicate a full stop sign? If the following sentence begins with a proper noun, not putting two periods may be confusing, in theory. Edit: It used to be 'contraction' instead of 'abbreviation'. How to do relative imports in Python? sir, problem about a host file sir, over department created a host file to run over office program in windows, but they give us dell system having operating system ubuntu 16.4 now my question is that how this add file in ubuntu to run the program. the windows command is given below  TO ACCESS FIORI - Please Follow the Instructions in this Video  Path:- C:\Windows\System32\drivers\etc  Copy attached Hosts File in the above path.  Fiori Link to access SAP in Google Chrome: Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$ Suppose that $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging the first two rows of $A$. Prove that $\det(B) = -\det(A)$.  By general $n \times n$ matrix, I mean $$\left[\begin{matrix} a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\ a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} \end{matrix} \right]$$ I'm thinking of using the cofactor expansion of the 2nd row of the first matrix, yielding $$ \det(A) = c_{21} a_{21} + c_{22} a_{22} + \cdots+ c_{2n} a_{2n}  $$ which will be positive or negative depending on $n$ and the values of $a_{21}, \ldots,a_{2n}$, and using the same on the first row in matrix $B$ to get $c_{11} a_{21}$ etc. and then use the fact that the power of the $c$ subscripts will alter $\det(B)$ to be the opposite sign, but I'm not sure how to express all of this mathematically in a proof directly. Help please? Evaluate $\lim\limits_{x \to a} \frac{x^m-a^m} {x-a}$ Multiple WHERE clause in Linq How to backup settings and list of installed packages If I want to make a backup of everything I have done since the fresh install of Ubuntu, what are the possible options? What all should I backup? I want to get all the settings that I changed, all the packages I installed, etc. Posting and answering questions you have already found the answer to I would like to know the physical meaning of the , if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates? How can I prevent inline math formulas from overflowing into the margin? When I compile my document in LaTeX, some formulas in the document in math mode, between $ $, go into the right margin area beyond all the text.  Is there a way to fix this aesthetic problem? How to create a site in a sitecollection on a remote SPServer using C# This is all in iTerm2 on OS X.  I used to have  export TERM='xterm-256color'   in my .bashrc. This meant Vim in tmux did use 256 colors.  And once I added  set -g xterm-keys on   then keyboard shortcuts with modifiers worked fine in Vim. Namely: shift+left/right that I mapped to switch Vim tabs, ctrl+up/down that I mapped to move ("bubble") lines, and shift+left/right which worked out of the box to jump by word in the command-line mode (e.g. when typing something like :e foo bar baz).  However, this setup had the problem that the Vim background color only shows behind text, as mentioned .  So I removed  export TERM='xterm-256color'   from my .bashrc and instead put this in my .tmux.conf:  set -g default-terminal "screen-256color"   That fixed the Vim background color, but broke the keyboard shortcuts - they do unexpected things (move the cursor, delete text) instead.  By using ctrl+v to insert the verbatim output from the key combinations (as described ), I was able to work around it:  map [1;5A &lt;C-Up&gt; map [1;5B &lt;C-Down&gt; map [1;2D &lt;S-Left&gt; map [1;2C &lt;S-Right&gt; cmap [1;2D &lt;S-Left&gt; cmap [1;2C &lt;S-Right&gt;   This makes the shortcuts work, but it doesn't feel like the right solution. Could anyone tell me what's happening here and how to fix it?