I have a trouble. I hope someone can do it.     How to prove that   $\left\{a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6} \mid (a,b,c,d)\in \mathbb{Q}^4\right\}$ is a field?   I can't prove the existence of multiplicative inverse. Thanks.
Why must $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ -- the set of all polynomials in $\sqrt{2}$ and $\sqrt{3}$ with rational coefficients  -- contain multiplicative inverses?   I have gathered that every element of $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ takes form $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{2}\sqrt{3}$ for some $a,b,c,d \in \mathbb{Q}$, and I have shown that $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ are both fields, but it's not clear to me why this allows for general multiplicative inverses to exist in $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$.
Can people PLEASE spell out their slang acronyms I am visiting Vietnam in November 2015, I will entering the country on the 10th and my flight home is 14 days later, I will be leaving Vietnam and travelling to Cambodia in between the 2 weeks then travel back into Vietnam to get my flight home on the 24th November, do I need a visa? Can't get hold of anyone at Embassy and not clear about entering twice on the website. Update help center article on search to reflect the new position of the search box The page  says:      Enter search terms in the search box that appears in the upper right corner of every page    But the box is not there. su vs sudo -s vs sudo -i vs sudo bash What is the difference between the following commands:  su sudo -s sudo -i sudo bash   I know for su I need to know the root password, and for sudo I have to be in the sudoers file, but once executed what is difference?  I know there is a difference between su and sudo -s because my home directory is /root after I execute su, but my home directory is still /home/myname after sudo -s. But I suspect this is just a symptom of an underlying difference that I'm missing. Can I update an early Airport Time Capsule? I have an Apple Time Capsule model A1254. It has a 1Tb drive. Is it possible to replace the drive with a larger capacity unit? It seems a shame to replace the whole unit. Thanks Trying to make aliases that open the last modified file One thing that I frequently do is edit the most recently modified files, so instead of typing "ls -lr" and then "vim lastfile", I thought I would make some shortcuts in my ~/.bash_profile file:  alias via="vim `ls -rt | tail -1`" alias vib="vim `ls -rt | tail -2 | head -1`" alias vic="vim `ls -rt | tail -3 | head -1`" alias vid="vim `ls -rt | tail -4 | head -1`" alias vie="vim `ls -rt | tail -5 | head -1`"   The problem is that, weirdly enough, these commands don't work.  They open some file that isn't one of the last, or even a file was deleted from the current directory (I wonder if there's some kind of file cache updating issue in the directory.  This occurs on both my local machine and the cluster I work on).  However, if I type vim `ls -rt | tail -1` directly, without using the alias, it works every time. Splitting of prime ideals in algebraic extensions Changing colors of lightning-button When running a Java app from eclipse my ImageIcon shows up just fine.  But after creating a jar the path to the image obviously gets screwed up.  Is there a way to extract an image from the jar at runtime so I can then open it up?  Or, is there a better way to do this?  I'd like to distribute a single jar file if possible. sudo: /etc/sudoers is owned by uid 501, should be 0 To run programs as sudo without password, I edited /etc/sudoers with wrong content.   Then I re-edited it with Mac's default text editor(course sudo cannot be used when /etc/sudoers is wrong).  I tried to run sudo agin, and get this error:   sudo: /etc/sudoers is owned by uid 501, should be 0 sudo: no valid sudoers sources found, quitting   What should I do? I cannot remember the root password (maybe I haven't ever set root password before). How do JavaScript closures work? How would you explain JavaScript closures to someone with a knowledge of the concepts they consist of (for example functions, variables and the like), but does not understand closures themselves?  I have seen  given on Wikipedia, but unfortunately it did not help. I need to calculate the following determinant in order to prove the following equality:  $$\det\begin{pmatrix} a &amp; b &amp; c &amp; d \\ -b &amp; a &amp; -d &amp; c \\ -c &amp; d &amp; a &amp; -b \\ -d &amp; -c &amp; b &amp; a \end{pmatrix} = (a^2+b^2+c^2+d^2)^2.$$  I tried using Gauss-algorithm to get an easier matrix, but I'm not sure if I did it correctly.  Calling the $4$ lines $I$, $II$, $III$ and $IV$, I did:  (1) $II \cdot a$  (2) $III \cdot a$  (3) $IV \cdot a$  After this I did:  (4) $II' + I \cdot b$  (5) $III' + I \cdot c$  (6) $IV' + I \cdot d$  So finally I got the following matrix:  $$\begin{pmatrix} a &amp; b &amp; c &amp; d \\ 0 &amp; a^2+b^2 &amp; bc-ad &amp; ac+bd \\ 0 &amp; ad+bc &amp; a^2+c^2 &amp; cd-ab \\ 0 &amp; bd-ac &amp; ab-cd &amp; a^2+d^2 \end{pmatrix}.$$  I thought this would make the determinant a bit easier, unfortunately I must have done something wrong. Is multiplication with single lines allowed as I have done it?  Thank you very much. iPad won't turn on after a complete battery drain My iPad mini retina is out of warranty,  I forgot to plug in my iPad after the critical battery warning and it went some minutes following the warning. I was not worried because same thing had happened a few times before and iPad was perfectly okay after charging it. However, this time I remember seeing a flash of blue screen immediately after it went black. Then, I plugged it in an waited for a while and tried starting it. Ever since, it fails to start hanging on the apple logo with white background screen.   I tried to connect to my Mac but nothing showed up in iTunes. I am worried now. Can I troubleshoot this or is it broken and needing repair? I see you have succussfully answered this question previously but it is not specific to my system.  I have downloaded your file and created the relevant dump file but I don't know how to analyse it. Can I upload it to you for some advice please? In node.tpl.php, I have done some changes, like creating an array of paragraphs:      $paragraphs = explode("&lt;p&gt;",render($content));   So, I have a for loop for showing the paragraphs, like:  $number = count($paragraphs); if($page){   print render($paragraphs[0]);   for ($i=1;$i&lt;$number;$i++)       {          print render($paragraphs[$i]);        &lt;a href="#commenting"&gt;Put comment for Paragraph &lt;?php echo $i?&gt;&lt;br&gt;&lt;/a&gt; &lt;?php  } ?&gt;     &lt;a id="commenting"&gt;&lt;?php print render($content['comments']); ?&gt;&lt;/a&gt; &lt;?php }?&gt;   As you can see I want to create a hyperlink under each paragraph, so when user click on it, they can put comment on the paragraph, But right now, when they click on the hyperlink, it will  jump to the end of the page, which has the comment form.  If you check the CommentPress website, in the right column, you can see for example, it is said: 5 Comment on Paragraph 1, and when you click on it, it will show the comments and the comment form under that link, I want to have the same thing, I mean, when users click on the link under each paragraph, the comment form displays under that link instead of jumping to the end of the page. Let $n\in\mathbb N$ and let $a_1,\ldots,a_n$ be natural numbers smaller than $10^n$. Write each $a_k$ in base $10$ and add $0$'s to the left of each decimal expansion, if needed, so that each $a_k$ is written with $n$ digits. Consider the matrix such that the entries of the $k$th row are the digits of $a_k$ (written in the same order). Let $d\in\mathbb N$ be such that $d$ divides each $a_k$. Prove that $d\mid\det A$.   For instance, suppose that $n=4$ and that your numbers are $3876$, $2784$, $684$, and $8388$, each of which is a multiple of $12$. Then$$A=\begin{bmatrix}3 &amp; 8 &amp; 7 &amp; 6 \\  2 &amp; 7 &amp; 8 &amp; 4 \\ 0 &amp; 6 &amp; 8 &amp; 4 \\  8 &amp; 3 &amp; 8 &amp; 8\end{bmatrix}$$and $\det A=-360$, which is, in fact, a multiple of $12$.  I learned about this problem yesterday. I found it quite cute and I decided to share it with all of you.  Note: I know how to prove it. Fastest converging method for calculating e Pronoun question: referring to inanimate objects as 'he' or 'she' I read the following claim concerning pronouns referring to inanimate objects:      Anything that is meant to contain you, protect you or provide you with something beneficial is [often referred to as] a she; anything that is a perceived threat is a he. That's why cars, boats and some countries are she.    Is this really the case, or is it just a subjective claim? (According to the  using he/she for inanimate objects is not recommended and it should be used instead.) Is there a 'rule' for determining whether to use he, she or it based on the impression one would like to express? (I only know the 'rule' for animals: In the case where you know their gender and they are important to you, you refer to them using he/she. E.g. A dog attacked me in the street. It bit me. versus This is my dog Roger. He is 7 years old.) Have these practices changed over time?   Note 1: There is already a similar question (), but that considers only the feminine case and none of the answers offers an objective discussion of the matter.  Note 2: My native language is Czech, where the gender of the pronoun is based only on the grammatical gender of the noun. E.g. a cat (kočka) is always she, a dog (pes) is always he, a boat (loď) is always she and a car (auto) is always it. I'm try to change to my Desktop directory from the terminal.  Running  cd /home/desktop   gives an error in the terminal saying no directory found.  What am I doing wrong? When editing a question on , I observed that a link which was not rendered in the post was hyperlinked in the edit preview (see pictures below). I tried reproducing it in the  to see if this also happens when the post is posted for the first time and observed the same. It seems to be a bug after the shift to CommonMark, leading to the different rendering of links on client and server-side. A link following a special character (without any space) is not hyperlinked in the post but is hyperlinked in the preview. A link separated from the special character with a white space appears to have no such problem. (see pictures below and my ). Link not hyperlinked in post  Same link hyperlinked in preview