I came up with a proof for this question, rather simple, that I didn't find here.  And I wondered if my proof was correct, and if anyone had a different proof ? Thanks for you time.  I will show that only polynomials  in $\mathbb{Q}[X]$ work.  My proof is based on induction on the degree of the polynomial,  For d = 0, it is obvious.  Suppose the result true for polynomials of degree $d$, and let P be a polynomial of degree $d+1$ that satisfies the problem,  $P(X) = a_0+\ldots+a_{d+1}X^{d+1}$.  Evaluating at 0, we find $a_0\in \mathbb{Q}$. Because $Q$ is stable by multiplication and addition, the polynomial $\dfrac{P-a_0}{X}$ is a polynomial of degree d that satisfies the constraints, the result follows by induction.
I'm studying abstract algebra for the first time.  let $F$ be a field. I learned that polynomials over field $F$ means polynomials with coefficients in $F$ (denoted by $F[x]$). But when I keep studying, I get to confused the word, 'polynomial functions from $F$ to $F$'.  Firstly, I consider this as induced functions from $F[x]$, so I guess all coefficients of these functions are in $F$. But let $F_{4}$ be a finite field consisting of four elements $\{0, 1, a, b\}$. and let $f: \{0,1\} \rightarrow \{0,1\}$ be a function defined by $$ f(x)=ax^2-ax$$ Since $\{0,1\}$ is a subfield of $F_{4}$ and $f(0)=0, f(1)=0$, I think $f$ is a example polynomial functions from $F$ to $F$ of which coefficients are not in $F$. Am I wrong? if so, please help me.  I encounter this problem while solving 'the set of all polynomial functions from $\Bbb{Z_{3}}$ to $\Bbb{Z_{3}}$ cannot possibly be isomorphic to $\Bbb{Z_{3}}[x]$'.
Itemize: Identation is different between first and second line How can I install MonoDevelop v4 on 12.04? I am using Ubuntu 12.04. I can install v2.8.6.3 of Monodevelop from software center but I need at least v3 to use Monogame v3 or higher. Can anyone point me in the right direction please?   Thanks Why is an orthogonal matrix called orthogonal? There are four options. Which of them is correct?- (a)${p\over k}$ (b)${k\over p}$ (c)$\log({p\over k})$ (d)${\log p\over k}$ I somehow want to use the rule for evaluating this kind of sum using integration, i.e. $\int_{0}^{1} f(x) \ dx=\lim_{n\to\infty} {1\over n}\sum_{r=1}^{n}f({r\over n})$  But the given expression $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]={1\over k}\lim_{n\to\infty}{1\over n}\sum_{r=0}^{p-k}\frac{1}{1+{r\over n}}$, sum running from $0$ to $p-k$ instead of $n$. So, how to evaluate this limit? Thanks for assistance in advance. Why, with Springer's  svmono template, do the letters written with the \mathcal font change? It seems that \mathcal becomes \mathscr.  Edit: Look at this image:    With the svmono template \mathcal returns the first alphabet and not the (classical) latter.  Here the packages I use in my document:  \documentclass[envcountsame,envcountchap]{svmono}  \usepackage{subfiles} \usepackage[utf8]{inputenc}  \usepackage{mathptmx} \usepackage[bottom]{footmisc} \usepackage{microtype} \usepackage{stmaryrd}    \usepackage{amsmath} \usepackage{courier} \usepackage{mathrsfs}  \usepackage{amssymb}  \usepackage[all,cmtip]{xy} \usepackage{faktor} \usepackage{graphicx} \usepackage{mathdots} \usepackage[colorlinks={true},linkcolor={blue},citecolor={blue}]{hyperref} \usepackage[frame,width=15truecm,height=23.5truecm,center]{crop}   With this settings I have the above strange behavior of \mathcal. I am designing a poster, which has some geometrical shapes (TikZ paths) surrounded by some text. I would like the text to flow neatly around these shapes. I bet there's some clever TikZ magic that could do this.  EDIT: This is the sort of effect I want, sorry to not be clear:    \documentclass[a4paper]{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \draw (0,0) to (2,-2) to (2,-4) to (4,-4) to [out=up, in=down] (6.5,0); \node [text width=6cm, anchor=north west] at (0,0) { \hspace{5pt}  This is the sort of effect I want to \\ \hspace{23pt} achieve, with text automatically \\ \hspace{36pt} constrained by some \\ \hspace{46pt} geometrical form. Here \\ \hspace{55pt} it's just a curve, \\ \hspace{57pt} but this could \\ \hspace{57pt} also be the \\ \hspace{57pt} text label \\ \hspace{57pt} of a node.}; \end{tikzpicture} \end{document} It's a bit difficult to implement a deep object copy function. What steps you take to ensure the original object and the cloned one share no reference? What are best practices for choosing and pairing fonts? I've always found font selection to be a little bit of a mystery. What factors should be considered when choosing a font? How does one successfully incorporate two fonts into a design?  I realize that some factors will include things like:   Legibility Scalability Appropriateness/Mood/Emotion   However, how does one objectively judge legibility or appropriateness? Are there objective principles that can be applied for font selection? If possible, please list resources or examples in your answers. Studies focused on the effectiveness of different fonts would also be extremely useful. When you send a group text message in iOS, it appears as if you automatically opt in to receive any and all replies to that message. Is there a way to avoid being included in group replies? Is there a way to remove oneself from a group chat? Removing all \n\r characters from a node XSLT? wondered if you could help me please? I have node in xml that is as followed  $LOG: 08880xbpnd $ fhdsafidsfsd df sd fsd f sd fsd  I was wondering is there anyway to make all the text go on to one line so that it then can be passsed through to a javascript function? so it would turn out like this   $LOG: 08880xbpnd $fhdsafidsfsddfsdfsdfsdfsd Reduce spacing between subequations I used to find color codes which I want to use for texts or background colors on htmls with color edit tool of MS paint which pre-installed on Windows. But now I use Mac instead of Windows. How can I find color codes with Mac? Is an software like MS paint pre-installed on Mac? Suppose there is a message that is encrypted with AES-128-CBC.  The message is as follows, new lines are used to delimit the 16 byte boundary for each block:  Wire funds from: Alice to Bob in   the amount of $ 1   Because this message is encrypted using CBC mode,  any modification of the first block of cipher text would propagate throughout the message.   However, the amount of the fund transfer falls in the last block.  I am under the assumption that an attacker can use the property of  to alter the last block and produce a valid amount that is greater than \$1 in far fewer than $2^{128}$ operations.   Is this a correct assumption?  Is there a name for this attack or do you know of a real world example?    My instinct is that if you just need one byte of the 16 byte block to be of a specific value then it would only require $2^8$ operations because the other 120 bits of the plain text could be any value. For example lets say we wanted the last block of cipher text to decrypt to 9\0 (where \0 denotes a single null byte, and the message would be interpreted as \$9), then it would take only $2^{16}$ operations to find this cipher text block.  Is there something I am missing? Why is everything floating in space so cold? Of course there is pretty hot stuff too. For example Suns.  But isn't the vacuum of space the perfect containment for heat? And shouldn't the rare collision with particles even heat up an object that floats around in space?  Why aren't they hot from time to time? Is it because they loose their heat by radiant heat?  And if so does radiant heat emission always occur until the object is the same temperature as the surrounding or does it depend on other factors? Mark's generosity in this crisis seems to more than make up for his earlier stinginess.   Should those sentences always be avoided, or are there cases where they are valid? How to prevent updating of a specific package? Opportunity attack when moving in and out of range of a creature in the same move? Let's say I'm 10ft away from an enemy. In my turn, I choose to move 5ft to get close enough to attack, I do attack, then I use my remaining movement to move out of range of the enemy. Does this still allow them an opportunity attack? Screen Resolution Problem with Ubuntu 14.04 and VirtualBox OpenID support for Ruby on Rails application What is current state of the art for enabling OpenID login in Ruby on Rails applications?  This is a community wiki with up-to-date answers to this question. Authlogic The most advanced authentication solution seems to be .  It supports OpenID with .  It supports Rails 4 and 3. Rails 2 is supported in the rails2 branch. You may want to watch  (and the ). There is a sample application called . Devise Devise is flexible authentication framework for Rails.  It supports OpenID with . restful_authentication Another authentication library of choice is  Rails plugin.  Seems like you also need to install  plugin. You may want to watch (old, circa 2007) . Ruby OpenID Raw support for OpenID protocol is handled by  library. Printing an array to a file with each element of the array in a new line in bash