Let $R$ be a field and consider $$A=\{n\in \mathbb N^*\mid \underbrace{1+...+1}_{n\ times}=0\}.$$ Assuming that it's not empty, prove that its smallest element is prime.  I have no idea how to do it. It looks here that we are in $\mathbb Z/n\mathbb Z$ and since $R$ is a field and that $A$ might be a subfield, then $n$ will be prime, but this don't answer the question since they ask for the smallest element.  If my question is unclear, let me know.
I'm trying to prove that the characteristic of any field $F$ is either $0$ or a prime number, but I have no idea what to do.  Help?
How do I draw text at an angle using python's PIL? Using Python I want to be able to draw text at different angles using PIL.  For example, imagine you were drawing the number around the face of a clock.  The number 3 would appear as expected whereas 12 would we drawn rotated counter-clockwise 90 degrees.  Therefore, I need to be able to draw many different strings at many different angles. Repeating an equation including its content and number I would like to re-use equations with the same content and number. I know that the thmtools package has a restatable environment to do this for environments (like in ), but I would also like to do this with a single equation that is only part of some environment.  I had a look at the following questions which only aim at repeating the equation number but typing the whole equation twice (the first three) or repeat the content of the equation without a number (the last one):         Based on the answers provided there and on the  I tried to create a macro \restatableeq for restable equations like this:  \documentclass{article}  \usepackage{amsmath}  \newcommand{\restatableeq}[3]{\label{#3}#2\xdef#1{\unexpanded#2\unexpanded\tag{\unexpanded\ref{#3}}}}  \begin{document}  \begin{eqnarray} \restatableeq{\eqone}{This &amp; is &amp; eqation 1}{eq1}\\ Another &amp; simple &amp; equation \end{eqnarray}  \begin{eqnarray} One &amp; more &amp; equation\\ \restatableeq{\eqtwo}{This &amp; is &amp; eqation 2}{eq2} \end{eqnarray}  Here, we use (\ref{eq1}) and (\ref{eq2}) again: \begin{eqnarray*} \eqone\\ \eqtwo \end{eqnarray*}  \end{document}   Unfortunately, it does not work and yields:  ! Missing { inserted. &lt;to be read again&gt;                    T l.10 ...ableeq{\eqone}{This &amp; is &amp; eqation 1}{eq1}                                                   \\ ?   My guess is that the &amp; symbols are not treated as intended but I may well be wrong there.  How can I define a command like \restatableeq that allows me to re-use equations with the same content and number in a setting like in this example (and give it a label such that I can still use \ref as usual in addition to that)? "How to [...]?" and "Where to [...]?" Questions that are not questions. Is this defensible? Adding a question mark to the end of a "How to" or "Where to" sentence appears to be quite common. Here are two examples from this very site:  How to punctuate a list of questions? ()  Where to put the periods when using a parenthetical sentence? ()  In my mind, the question mark doesn't belong at all, and it bothers me.  Am I being unreasonable? Is this practice defensible? Italian keyboard: entering tilde (~) and backtick (`) characters without changing keyboard layout Here is what the layout of an Italian keyboard looks like:  As you can see, the curly braces, which are absolutely necessary to write code in any of C, C++, JavaScript, Java, Objective-C, and others, appear in purple in this picture. This is because while the other characters can be seen visually as printed characters on most Italian keyboards, these purple characters are actually usually not printed on the keyboard's keys. These characters, found in the upper right corners of the given keys, can usually be generated with the following key combinations: (RIGHT SHIFT or LEFT SHIFT) + (RIGHT CTRL or LEFT CTRL) + (Alt or AltGR) + ([ or ]) (Actually as the MSKLC application shows, SHIFT + AltGr + [ and SHIFT + AltGr + ] are enough to generate the curly braces.) Characters such as the at (@) sign (called 'chiocciola' in Italian) and the hash (#) key (called 'cancelletto' in Italian) found in the lower right hand corner of the keyboard can be generated easily as follows (AltGr + @) or (AltGr + #). Such characters are also fundamental to programming as they can, for instance, be found in CSS, namely with media queries and with selectors referring to the id html attribute. Upon further inspection, we can see that all of the US ASCII keyboard / programmer's keyboard are covered by the keys of the Italian keyboard, except for the following, which are the tilde, which is especially useful on the Linux command line and is used to refer to a user's home directory on such system, as well as the back-tick (`), also known as the back-quote, which is also very useful on the Linux shell when we want to turn the output of a command into a command-line input. So, here comes the question: With all those empty upper-right hand corners of keys, why isn't there a better way of generating these characters, perhaps through a CTRL-ALT-SHIFT modifier key combination? All I've been able to find regarding generation of these characters are the following ALT-codes:  ALT code 96  corresponds to ` ALT code 123 corresponds to { ALT code 125 corresponds to } ALT code 126 corresponds to ~  and to generate a character via an ALT code, you make sure that the NumLock key on your keypad is toggled on, hold down the ALT key, enter the keycode, and release the ALT key. ALT code are documented at:  If someone knows of a better way of inserting these characters from an Italian keyboard without switching the keyboard layout in the Control Panel, then please let me know. Thanks. How to create two classes in C++ which use each other as data? How to comment lines in rails html.erb files? In older versions of Windows and even in Windows 10 previews you could enable the built-in guest account in control panel     Control Panel\User Accounts\User Accounts\Manage Accounts     In Windows 10, the Guest entry is missing.  Using Computer Management or net user you can still enable the guest account, but even then it doesn't show up on the logon screen.  In the Local Security Policy secpol.msc:     Local Policies - User Rights Assignment - Deny log on locally   we can see guest listed, this wasn't the case in older versions of Windows.  After removing guest from that policy, it is listed on the logon screen and we can using it to log on.  However, the whole Explorer shell is crashing and restarting constantly. The desktop is blinking and you can't click anything.  So the guest account is totally broken.   There is a  about this showing that some code was quickly added to the control panel to not show the guest account. So Microsoft does know about the problem.  While I don't really expect anybody to know what they've done to break this, at least this explains why the guest account is currently not listed. I am running Ubuntu 16.10 on a MacbookPro and keys 49 and 94 are inverted... I resolved this problem with xmodmap, but this won't apply permanently. After some research I found xmodmap is no longuer use by Ubuntu but xkb instead. So I searched in the configuration of xkb and I found in the config file /usr/share/X11/xkb/keycodes/macintosh this lines :   xkb_keycodes "badmap" {     &lt;TLDE&gt; =  94;     &lt;LSGT&gt; =  49; };  xkb_keycodes "goodmap" {     &lt;TLDE&gt; =  49;     &lt;LSGT&gt; =  94; };   This is exactly my problem, this two keys are inverted. So I wondering if I can't change my layout configuration with the param badmap.    The ouput of setxkbmap -query -v 10  Setting verbose level to 10 locale is C Trying to load rules file ./rules/evdev... Trying to load rules file /usr/share/X11/xkb/rules/evdev... Success. Applied rules from evdev: rules:      evdev model:      pc105 layout:     fr,fr,us variant:    mac,oss, Trying to build keymap using the following components: keycodes:   evdev+aliases(azerty) types:      complete compat:     complete symbols:    pc+fr(mac)+fr(oss):2+us:3+inet(evdev) geometry:   pc(pc105) rules:      evdev model:      pc105 layout:     fr,fr,us variant:    mac,oss, Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ideas  like $$S(x)+S'(x)+S''(x)=e^x-1\,.$$Maybe if I get the third equation it will allow me to construct a kind of differential equation. Then, by solving it, obtain $S(x)$. QGIS Processing - Additional Creation Parameter for GdalWarp I am having a hard time getting the "Additional Creation Parameters" to work when using GdalWarp through the processing toolbox.  I am trying to set the nodata value to 0 for the output raster.  processing.runalg("gdalogr:warpreproject","InRaster.tif","EPSG:32736","EPSG:32736",10,0,"-dstnodata 0","OutRaster.tif")   Has anyone used this functionality and managed to get it to work? I get this warning:     Java HotSpot(TM) 64-Bit Server VM warning: Using incremental CMS is deprecated and will likely be removed in a future release   What does it mean? Can HIV be transmitted via mosquitos? It is known that HIV is usually transmitted by direct blood or body fluid contact between an infected individual and a healthy person (like blood transfusion or needle sharing):  Suppose a mosquito bites an individual suffering from AIDS and in the process sucks up some T cells infected with HIV along with RBCs. Then it bites another person not suffering from the disease, and transfers these infected T cells. Isn't there a high probability of the second individual contracting HIV? Who is the Community user? Dimension of cone of projective variety How much time have the Straw Hats spent with each other? Is there any event that fires when WPF animation ends? Why does a query run slower in a Stored Procedure than in the Query window? I have some auto-generated T-SQL, which is probably valid, but I don't really understand.  ALTER TABLE [dbo].[MyTable] WITH CHECK CHECK CONSTRAINT [My_FORIEGN_KEY];   I know what a foreign key constraint is, but what's the CHECK CHECK? How to change from csh to bash as default shell I have csh as my default shell, as shown by echo $SHELL. I want to switch to bash as my default shell. I tried the following approaches to no avail:   With chsh I get:  chsh: can only change local entries; use ypchsh instead.  With ypchsh I get:  ypchsh: yppasswdd not running on NIS master host ("dcsun2").    I only have .chsrc in my home directory and I cannot find any .profile files in /etc. How can I change my default shell to bash? Which Stack Exchange website for machine learning and computational algorithms? I'm currently working on many machine learning and computational algorithms, such as Singular Value Decomposition, Support Vector Machines, and others.  I'd like to ask questions about these topics, but I don't know which Stack Exchange website to use to post my questions.   Should I use ? Should I use ? Should I use ? Should I use ?   How do I tell?  What criteria do I use?