Let $K$ be a field. I want to show that $(X^2-Y^3)\subset K[X,Y]$ is a prime ideal.   I know that $K[X,Y]$ is factorial, so it is enough to show that $X^2-Y^3$ is irreducible. But Eisenstein does not to work here, does it?  How can I see that $(X^2-Y^3)$ is prime ideal?  Thanks in advance.  Edit: The task is to show that $K[X,Y]/(X^2-Y^3)$ is a domain. I thought it is best to show that $(X^2-Y^3)$ is a prime ideal in $K[X,Y]$. Am I mistaken?
How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime?    I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel is exactly $I$. But the difficulty is that how to prove the kernel is contained in $I$.  Could you help me? Thanks a lot!
I want to know a proof of an alternative form of Fermat-Euler's theorem   $$a^{\phi (n) +1} \equiv a \pmod n$$   when $a$ and $n$ are not relatively prime.   I searched some number theory books and a cryptography book and internet, but there were only proofs of the original theorem  $a^{\phi (n)} \equiv 1 \pmod n$ or in case n=pq which is used for RSA. So I would be very thankful if someone show me a proof or book I should read. How to create a label combining different font sizes or types? I would like to place the elevation-number of a point in a different front size and centred under its name:    Is that possible?  (That is my actual labeling: label || '\n' || elevation) I would like to use pgfplots to create a figure where I plot curves that have a recursive formula. I tried a very simple case that uses the declare function command to see whether this is possible, and I can't compile (TeX capacity exceeded). Is it even possible to do this?  \documentclass{standalone}  \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture}[   declare function={test(\p)=ifthenelse(\p&gt;1, test(\p-1), 0.5);}, ]     \begin{axis}[]     % use TeX as calculator:     \addplot [mark=none] {x};     \addplot [mark=none] {test(0)};     \end{axis} \end{tikzpicture}  \end{document} As far as I know, those are the only two.  They should be pronounced Soogher and Soor, shouldn't they?  I looked them up on Dictionary.com, and their etymologies reveal no trace of an SH, except where the listing for sugar had:     Middle English sugre, sucre (noun) &lt; Middle French sucre  &lt; Medieval Latin succārum  &lt; Italian zucchero  &lt; Arabic sukkar;  obscurely akin to Persian shakar, Greek sákcharon   I see an obscure kinship to shakar, but the word morphed so many times since then that the SH disappeared pretty much completely.  And sure is even worse, with no sign of SH:     Middle English sur ( e ) &lt; Middle French sur, Old French seur  &lt; Latin sēcūrus   Why are these the only two like this? Big tilde in math mode How do I get a really wide tilde?  I need to cover at least  abcdefghijklmnopqrstuvwxyz  The \widetilde command does not work. Nor using \stackrel with \sim. I've just updated to the latest Google Chrome Dev build, and they've changed from the annoying-but-ok new tab page to an ugly and less usable page.  On the old one, nine thumbnails was not enough, and now they've restricted it to eight.  There's an empty "Tips and Suggestions" box which I wouldn't want even if it had stuff in; they've added Recent Downloads - on the rare occasions when I need that, I'll hit Ctrl-J to open the Downloads tab; and the site-specific search box on the right has now gone!  Is there a way to get the old new tab page back, or a plugin to provide a superior one?   Update: I launch Chrome by pressing Win+R and entering "chrome" - I have a shortcut on the Path with a Target of "C:\Documents and Settings\&lt;user&gt;\Local Settings\Application Data\Google\Chrome\Application\chrome.exe" --old-new-tab-page and whilst double-clicking this shortcut brings up the previous new tab page, using the keyboard method results in the old one.  How do I get it working via the Run box? (Without using a batch file, which slows things down too much.)  Update 2: GRRR! Now they've gone and stopped the --old-new-tab-page parameter from working at all! I hate this new version of the page that they're forcing upon us - can anyone provide a method to get rid of it and get the decent page back? Explaining to laypeople why bootstrapping works I recently used bootstrapping to estimate confidence intervals for a project. Someone who doesn't know much about statistics recently asked me to explain why bootstrapping works, i.e., why is it that resampling the same sample over and over gives good results. I realized that although I'd spent a lot of time understanding how to use it, I don't really understand why bootstrapping works.  Specifically: if we are resampling from our sample, how is it that we are learning something about the population rather than only about the sample? There seems to be a leap there which is somewhat counter-intuitive.  I have found a few answers to this question here which I half-understand. Particularly . I am a "consumer" of statistics, not a statistician, and I work with people who know much less about statistics than I do. So, can someone explain, with a minimum of references to theorems, etc., the basic reasoning behind the bootstrap? That is, if you had to explain it to your neighbor, what would you say? How to merge cells vertically Is it possible to create a table like this one?    The usual structure of tables I use looks like this:  \begin{table}[H]     \begin{adjustbox}{width=\textwidth,center}     % \begin{adjustbox}{center}         \begin{tabular}{lll}             \hline             \multicolumn{1}{l}{\head{XXX}} &amp; \multicolumn{1}{l}{\head{XXX}} &amp; \multicolumn{1}{l}{\head{XXX}} \\             \hline             \ttfamily xxx &amp; \ttfamily xxx &amp; \ttfamily xxx \\             \hline         \end{tabular}     \end{adjustbox}     \vspace{ - 05 mm}     \caption{xxx}     \label{tab:xxx} \end{table} Configuring, compiling and installing a custom Linux kernel How to draw density estimate on histogram by pgfplots? I know how to draw a density histogram using pgfplots from this answer: .  But how can I draw an estimate line on the histogram like this?    \documentclass{article} \usepackage{pgfplots} \begin{document} \pgfplotsset{     small,     every axis plot post/.style={         fill=orange!75,         draw=orange!50!black     },     trim axis left }  \begin{tikzpicture} \begin{axis}[ymin=0, title=\texttt{hist=density}] \addplot [hist=density] table [y index=0] {random.txt}; \end{axis} \end{tikzpicture} \end{document} How can I produce the text Word_one_two in LaTeX?  I tried:  Samp\_Dist\_Corr   But, it doesn't quite look right. Also, I want it in the typewriter font, so actually, I'm doing:  \texttt{Samp\_Dist\_Corr}   I find it looks a bit like the underscore is merging in to the bottom of the "D", but maybe it's just because of the typewriter D? Makeindex error when ! used in \index in align environment See MWE below. The symbol ! in the \index causes an error. This is only the case in align environments (not in equation). So it seems to have to do something with amsmath. If ! is put in an \mbox there is no error. But then of course the font size is not adjusted to subindices. Curiously enough, \index{$C_{K!!}$} (two exclamation marks) doesn't cause any error.    \documentclass{article} \usepackage{amsmath} \usepackage{imakeidx} \makeindex \begin{document} \begin{align} y\index{$C_{K!}$} \end{align} \printindex \end{document}     Any solutions suggested? Drawing arrows on an HTML page to visualize semantic links between textual spans I have an HTML page with some textual spans marked up something like this:  ... &lt;span id="T2" class="Protein"&gt;p50&lt;/span&gt; ... &lt;span id="T3" class="Protein"&gt;p65&lt;/span&gt; ... &lt;span id="T34" ids="T2 T3" class="Positive_regulation"&gt;recruitment&lt;/span&gt; ...   I.e. each span has an ID and refers to zero or more spans via their IDs.  I would like to visualize these references as arrows.  Two questions:   How can I map an ID of a span to the screen coordinates of the rendering of the span? How do I draw arrows going from one rendering to another?   The solution should work in Firefox, working in other browsers is a plus but not really necessary. The solution could use jQuery, or some other lightweight JavaScript library. How can timelapse sequences be shot without in-camera support? How can timelapse sequences be shot without having in-camera support for it?  Which third-party tools are recommended? How can I get an underbrace and an overbrace to partially overlap in an equation? Sometimes I want to describe portions of an equation using \underbrace and \overbrace in such a way that the portions are not completely distinct, for example:          /-------\ x = A + B + C + D + E     \-------/   \---/      What is the best way to do this? "debconf: DbDriver "config": config.dat is locked by another process: Resource temporarily unavailable" while installing packages Convergence of a sequence implies convergence of the norm of the sequence If $\langle X_n \rangle$ is a sequence in $\mathbb R^n$ then $\lim\limits_{n\to \infty}X_n=A$. Show that $\lim\limits_{n\to \infty} \| X_n \| = \| A \|$ .   My thought process, $\| X_i \|= (\sum_{i=1}^n x_i^2)^{(1/2)}$ therefore be the multiplicative property of limits $x_i^2 = a_i^2$. Therefore the limit of norm of the components is equivalent to the limit of the corresponding limit of each component of the sequence. How should you interview for QA *Automation* positions? Can a possessive S be attached to nouns that are not human beings (or animals)? Can a possessive S be attached to nouns that are not human beings (or animals)?  For example, instead of saying "the back of the chair", can I say "the chair's back"?   I remember learning that not everything can get a possessive s, but this issue is not clear to me. Can I share my usb keyboard between two computers I love my Filco keyboard, I use it as primary keyboard for my Lenovo thinkpad all time. Now Im moving into the mac world and wonder:  is it possible to split my USB keyboard between two USB ports, ie one at my lenovo and one at my new iMac.   I was thinking that maybe there exists some kind of box with a switch to decide which port should receive the input, or preferable it would be possible to switch between the two via a keyboard shortcut.