Let $L$ and $K$ number fields, if $p$ is ramified in $LK$ then it's ramified at $L$ or $K$?
If $L_i/K$ are Galois extensions of number fields, $i=1,\ldots,n$, and $L=L_1\cdots L_n$ is the compositum. Then it's true that a prime $\mathfrak{p}$ of $K$ splits in $L$ if and only if it splits in all of $L_i$. Does this also hold if the $L_i$ are not Galois extensions of $K$?  The proof that I know regarding the compositum uses the fact that the $L_i/K$ are Galois, so is this true in the more general setting and how would one prove it?
How do I install a fingerprint reader on Lenovo ThinkPad? Can I get tourist visa on arrival at Dubai airport? I don't understand why su - is preferred over su to login as root. How to add current date to cell if the rest of the cells (or some cells) in that row are edited, but only the first time? I have a column A4:A (heading = Date), and other columns after it B4:O.  I want column A to fill in today's date automatically when I fill in any other columns (some of these B4:O maybe empty, but most will be filled) in that row. But I only want that to happen the first time I edit any other columns within that row.  What I've tried:  A7: =IF(B7 &gt; 0,TODAY(),) Windows Managers vs Login Managers Vs Display Managers Vs Desktop Environment I posted a  and noticed people weren't distinguishing correctly between many of these things: Windows Managers vs Login Managers Vs Display Managers Vs Desktop Environment.  Can someone please clear this up, i.e. tell us the difference between them and how they are related perhaps?  What category does Xorg fall under?  What about Gdm/Kdm/Xdm?  People also talk about X.  What is X? More examples of Simpson's Paradox, barring the ones on Wikipedia, Titanic, and delayed flights. I hope someone may know fresh examples of  for use in my statistics courses.  The examples I've been using are fine, but I'd like to have some new ones.  I'm familiar with the ones on the Wikipedia page: the gender bias lawsuit at Berkeley, batting averages, mortality rates among low birth babies, party voting on the Civil Rights Act, and success rates of kidney stone treatments.  I'm also familiar with the example of survival rates on the Titanic and the one about delayed flights at America West vs. Alaska Airlines.  These are all good ones, but, like I said, I'm looking for more. Does anybody know any others?  For those who aren't familiar with it, Simpson's paradox is essentially a property of unreduced fractions: It's possible to have, simultaneously, $$\frac{a_1}{b_1} &lt; \frac{c_1}{d_1} \text{ and } \frac{a_2}{b_2} &lt; \frac{c_2}{d_2}, \text{ but } \frac{a_1 + a_2}{b_1+b_2} &gt; \frac{c_1+c_2}{d_1+d_2}.$$   For example, $\frac{1}{3} &lt; \frac{34}{100}$ and $\frac{66}{100} &lt; \frac{2}{3}$, but $\frac{67}{103} &gt; \frac{36}{103}$.  It can be hard to spot this happening in a given real-world scenario. Do I need a transit vsa in Kuwait; what about checked baggage? Gravitational potential at a point in a gravitational field is defined as work done per unit mass in bringing a small test mass from infinity to the point.  I know that:   By integration, we find that gravitation potential is negative. The maximum value of gravitational potential which is at infinity distance is 0, therefore minimum value should be negative.   But, what I do not understand is force is in the same direction as displacement so why work done is not positive? Using a question mark mid-sentence latex table border dont appear / to big / to small still working on that table. Here's my new question reguarding the table borders:  It appears, that i do something wrong setting boarders. Here is my code, and a screenshot:  \documentclass{article} \usepackage{multirow} % Used for the vertical cells \usepackage{rotating} % Used for the vertical cells  \usepackage{xcolor} % Used for cell-colors \usepackage{colortbl} % Used for cell-colors  \usepackage[a4paper]{geometry}% Used to set the pdf margins manually \newgeometry{left=2cm,bottom=0.1cm}  \definecolor{grey}{HTML}{F2F2F2} % Define grey for the cells \definecolor{red}{HTML}{FF0000} % Define red for testing  \begin{document} \scriptsize % Use a small font size  \begin{tabular}{|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0,6cm}|}  \cline{10-17}  \multicolumn{9}{|c|}{} &amp; \multicolumn{2}{p{\dimexpr (0.6cm)*2+2\arrayrulewidth+2\tabcolsep\relax}|}{\cellcolor{grey}Work to retired ratio} &amp; \multicolumn{4}{p{\dimexpr (0.6cm)*4+1\arrayrulewidth+2\tabcolsep\relax}|}{Health care sizing (per thousand)} &amp; \multicolumn{2}{p{\dimexpr (0.6cm)*2+1\arrayrulewidth+2\tabcolsep\relax}|}{\cellcolor{grey}Inverse dependency ratio}\\\cline{3-9}  \multicolumn{2}{|c|}{} &amp; \multicolumn{4}{c|}{\cellcolor{grey}Population (mio.)} &amp; \multicolumn{3}{c}{Workforce (1000)}  &amp; \multicolumn{2}{|c|}{\cellcolor{grey}} &amp; \multicolumn{2}{l|}{Physicians by} &amp; \multicolumn{2}{l}{Nurses by} &amp; \multicolumn{2}{|l|}{\cellcolor{grey}Old agers by}\\\hline  \multicolumn{2}{|c|}{} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Overall}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Young age}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Overall}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Physicians (Proj: 2025)}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Nurses (Proj: 2030)}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}Working age by Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}Work force by Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Workforce}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Workforce}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}1 physicians}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}1 nurse}\end{sideways}\\\hline  \multicolumn{2}{|l}{USA} &amp; \multicolumn{4}{l}{\cellcolor{grey}} &amp; \multicolumn{3}{l}{} &amp; \multicolumn{2}{l}{\cellcolor{grey}} &amp; \multicolumn{4}{l}{} &amp; \multicolumn{2}{l|}{\cellcolor{grey}}\\\hline  1990/1 &amp; &amp; 248,6 &amp; 71,3 &amp; 146,1 &amp; 31,2 &amp; 127.500 &amp; 615,4 &amp; 1491,6 &amp; 4,68 &amp; 4,09 &amp; 4,83 &amp; 4,21 &amp; 11,70 &amp; 10,21 &amp; 50,7 &amp; 20,9\\\hline  1 &amp; 2 &amp; 3 \\\hline  \end{tabular}  \end{document}     On 1 the Top border of Population is not visible although i did used \\\cline{3-9}\cline{12-17} on that line.  In 2 both borders are pushed a bit to the right for no reason i can see.  And in 3 again the top borders are not visible despite using \cline{10-17} on that line. What did i do wrong?  Thanks! Let, $X$ be a countable set. Which of the following are true?   There exists metric $d$ on $X$ so that $(X,d)$ is complete. There exists metric $d$ on $X$ so that $(X,d)$ is not complete. There exists metric $d$ on $X$ so that $(X,d)$ is compact. There exists metric $d$ on $X$ so that $(X,d)$ is not compact.   4 is true as we can take discrete metric. Then $\{\{x\}:x \in X\}$ is an open cover containing no finite subcover.  If there exists a compact metric then it would automatically become complete also. Hence if 3 is true, so is 1.  What about 2? Any other comment? Auto comment "possible duplicate of ..." missing on voting to close as duplicate How to use siunitx and tabularx together? I'm using the tabular features of siunitx (for what it is worth: v1.3 since I'm on TexLive 2009) to align numbers in tables, eg.  \begin{tabular}{ p{1.5cm} S S S } bla &amp; 1.23 &amp; 4.5 &amp; 67.89 \tabularnewline \end{tabular}   and that works perfectly fine.  Now I would like to have this table span a specific width, or rather to have several tables to have the same width. For this I would normally use tabularx  \begin{tabularx}{\textwidth}{ p{1.5cm} X X X } bla &amp; 1.23 &amp; 4.5 &amp; 67.89 \tabularnewline \end{tabular}   Now, is there a way to combine the two? To have the number formatting and alignment features of siunitx inside an automatically resizing table? Validity of naively computing the de Broglie wavelength of a macroscopic object Many introductory quantum mechanics textbooks include simple exercises on computing the de Broglie wavelength of macroscopic objects, often contrasting the results with that of a proton, etc.  For instance, this example, taken from a textbook:     Calculate the de Broglie wavelength for      (a) a proton of kinetic energy 70 MeV      (b) a 100 g bullet moving at 900 m/s   The pedagogical motivation behind these questions is obvious: when the de Broglie wavelength is small compared to the size of an object, the wave behaviour of the object is undetectable.  But what is the validity of the actual approach of applying the formula naively? Of course, a 100g bullet is not a fundamental object of 100g but a lattice of some $10^{23}$ atoms bound by the electromagnetic force. But is the naive answer close to the actual one (i.e. within an order of magnitude or two)? How does one even calculate the de Broglie wavelength for a many body system accurately? Importing coordinate data from Excel into QGIS I have an Excel file containing data collected over many years as latitude and longitude coordinates for archaeological sites. How can I import it into QGIS as a new layer? I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$.  I tried applying the Mean Value theorem like this: There exists a $c \in (a, x)$ st $f(x)-f(a)=f'(c)(x-a) \implies \frac{f(x)-f(a)}{x-a}=f'(c) \implies f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{c\to a}f'(c)=A$  However, I think this is wrong. I am pretty sure the Mean Value theorem is needed, but my application of it seems to be incorrect. Would somebody be able to point me in the right direction? Why is $2\pi i \neq 0?$ We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$  Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's formula), then shouldn't $2\pi i=0$? What am I missing here? How create Makeself self executable archive file in LINUX? Is it possible to paginate posts correctly that are random ordered? I found this problem on Wordpress Support and the topic unfortunately is now closed.  I have this same issue... (read below)    We have created a site where members can recommend things like favourite books, movie, songs, etc. For this problem I will use the Movies page as an example.  The "Movies" page is ultimately a custom page template that asks wordpress to display a random list of ALL posts that have been given the category "movies" (category 31). It displays the title of these movies in random order using the code below.  &lt;?php  $rand = new WP_Query("cat=31&amp;showposts=-1&amp;orderby=rand");  while($rand-&gt;have_posts()) : $rand-&gt;the_post(); ?&gt;" rel="bookmark" title="Permanent Link to &lt;?php the_title_attribute(); ?&gt;"&gt;&lt;?php the_title(); ?&gt; &lt;?php endwhile; ?&gt;   Problem is the list is getting rather long and I would like to split it up into two or more pages of about 10 movies each. To achieve this, I have used the code below.  &lt;?php  $page = (get_query_var('paged')) ? get_query_var('paged') : 1;  query_posts("cat=31&amp;orderby=rand&amp;showposts=10&amp;paged=$page");  while ( have_posts() ) : the_post()  ?&gt;" rel="bookmark" title="Permanent Link to &lt;?php the_title_attribute(); ?&gt;"&gt;&lt;?php the_title(); ?&gt; &lt;?php endwhile; ?&gt;   But there is a problem because although it splits up the data into pages of 10 posts each (paginates), it fails to include a new set of 10 posts on page 2 and so forth. In other words, because it is listing things in a random order, it just goes out and gets another 10 random posts (or in this case movie titles). As a result, we have some repeated movie title posts instead of a new set of 10 random movie titles on page 2, etc.  My question is - what can I add to this code to get wordpress to "remember" which random 10 posts it included on page 1, and then have it get a new set of 10 posts to put on pages 2, 3, etc. until all posts are displayed. I would like there to be only one occurrence of a post per page when sorted randomly in sets of 10. How do I check whether a file exists without exceptions? How do I check whether a file exists or not, without using the  statement?