Is a left and right simple ring with unity a division ring?
Division ring if and only if it has no proper left ideals.
Why must a field whose a group of units is cyclic be finite? Let $F$ be a field and $F^\times$ be its group of units. If $F^\times$ is cyclic, then show that $F$ is finite.  I'm a bit stuck. I know that I can represent $F^\times = \langle u \rangle$ for some $u \in F^\times$ and that we must have that $|F^\times| = o(u)$, where $o(u)$ denotes the order of $u$ in $F^\times$.  I tried assuming $o(u) = \infty$, but I'm not sure exactly where to go from there. I was wondering if I could get a hint. Let $X(n)$ denote number of all permutations of $\left\{1,\ldots,n \right\}$ that have only cycles of even length. Let $Y(n)$ denote number of all permutations of the same set that have only cycles of odd length. Count $X(2n)-Y(2n)$.   I don't know even how to start. I counted this sum for $n=1,2,3$ and it seems that it is always equal to $0$, but it doesn't help with calculating it in general. Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$ How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$  Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$.  I would however like to learn the technique of dealing with surds such as these, especially in trignometric problems as I have a lot of similar problems and I don't have a clue as to how to deal with those.  Hints please!  EDIT:  What I have done using half angles is this: (and please note, for convenience, I am dropping the degree symbols. The angles here are in degrees however).  I know that $$ \cos 15 = \dfrac{\sqrt3+1}{2\sqrt2}$$  So,   $$\sin7.5 = \sqrt{\dfrac{1-\cos 15} {2}}$$ $$\cos7.5 = \sqrt{\dfrac{1+\cos 15} {2}} $$  $$\implies \cot 7.5 = \sqrt{\dfrac{2\sqrt2 + \sqrt3 + 1} {2\sqrt2 - \sqrt3 + 1}} $$ How to view recently opened media files? In windows we have an option to view recently opened media files as my recent documents, I was  wondering does there exists something like that in Ubuntu?  via terminal or via GUI? Undefined variable problem with PHP function How do I add a custom script on a specific page? I wanted to add custom JavaScript code on certain nodes. I have written this code in the html.tpl.php template, but this doesn't work. (donate is the path alias for a node.) &lt;?php if(arg(0) == 'DONATE'): ?&gt;   &lt;script&gt;     (function() {       // ...     );   &lt;/script&gt; &lt;?php endif; ?&gt; What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is arbitrary. So saying $\sqrt{-1} = i$ is plainly false!  So why do people say $\sqrt{-1} = i$? Is this how it's taught in the US? In a custom lightning component where one is using the recordEditForm tag, how can one validate fields that are filled by the user?  For example, in a quick action to create a Case record, we can have the following line inside the recordEditForm:  &lt;lightning:inputField aura:id="caseAccount" fieldName="AccountId"/&gt;   How can one enable or disable a button if certain fields are assigned by the user? Like if the account is specified, then a button on the component is enabled for the user.  Making the field required is not an option. Clear some doubts and questions about the Witch-king of Angmar I recently found out about the war between Arnor and Angmar, and was pretty shocked to learn that the Witch-king basically wiped the floor with the Dúnedain kingdom... I mean he's basically the sole reason Aragorn wanders around like a homeless person.  So with which army did the Witch-king defeat Arnor? Orcs? Men? Sorcery? Did he ever go to battle himself? Did he do this for laughs or by Sauron's orders?   How powerful was the Witch-king at this point, compared to in the Fellowship of the Ring and Return of the King? Were the other Nazgul with him in this war?   And it doesn't even matter to me if all of the eight Nazgul were with him, even if they all had their Rings of Power, and even if they had the largest of armies of Orcs, Men, Elves, Dwarves or even dragons (yes I'm exaggerating at this point), but just how do you defeat a kingdom full of Dúnedain?   We have all seen what Aragorn is capable of doing, so how do you defeat a kingdom worth of Aragorns?   And if it is not much to ask, does anyone know where I can read all about this war and more about the Witch-king? so far all I read about it was from the Internet, wikis, forums and alike, I'd like to read it from the source. This is a general, community wiki  question to address non-specific "I need a CMS or Wiki that does x, y, and z..." questions.  If your question was closed as a duplicate of this question and you feel that the information provided here does not provide a sufficient answer, please open a discussion on .    I have a list of features that I want for my website's Content Management System (CMS) - where can I find a [free] script that includes all of them? If you have an $m \times n$ matrix and the rows are linearly independent, are the columns linearly independent too?  I know that it's the case for $n \times n$ matrices. I've yet tried to think about it, and I think the answer is yes, for example the matrix: $\begin{bmatrix} 1 &amp; 2 &amp; 3 \\ a &amp; 2a &amp; 3c \end{bmatrix}$ has linearly dependent rows and columns, but I'm not sure that it works for every $m \times n$ matrix. What kind of node is this? I have this table definition as part of a invoice  \begin{tabularx}{\textwidth}[ht]{lXrr}%   It worked fine for years until I have a very short word in the X column for the first time. Now my table is very small in width and centered horizontally on the page.  In my expectation the X column should fill up all space until the width of the table reaches the width of the page except the margins.  I have read about workarounds specifying something like 1.2\textwidth, but I don't like that, because the table is a template included via \include, so it would affect all my invoices.  I also tried \linewidth, \hsize and \columnwidth, but everything results in the same layout.  Is there a way to make the only X column expand so that it uses all available space?    Complete included table definition (Rechnungstabelle.tex):  \newcommand{\Rechnungstabelle}[4]{% vielen Dank für Ihren Auftrag. \noindent \begin{tabularx}{\textwidth}[ht]{lXrr}% \toprule% \bf{Anzahl} &amp; \bf{Bezeichnung} &amp; \bf{Einzelpreis} &amp; \bf{Gesamtpreis} \\% \midrule% #1% \bottomrule% &amp; \bf{Rechnungsbetrag}  &amp; &amp; \bf{\EUR{#2}} \\% \ifthenelse{\equal{#3}{}}{}{&amp; Enthaltene MwSt. 19\% &amp; &amp; \EUR{#3} \\}% \ifthenelse{\equal{#4}{}}{}{&amp; * Enthaltene MwSt. 7\% &amp; &amp; \EUR{#4} \\}% \end{tabularx}% }   Changing the table header row to  \bf{Anzahl} &amp; \bf{Bezeichnung} \hspace{\textwidth} &amp; \bf{Einzelpreis} &amp; \bf{Gesamtpreis} \\%   is a workaround, but I don't think that would be good TeX style.  Minimum example for reproducing:  \documentclass[ fontsize=12pt,                  paper=a4,                 foldmarks=TBMPL,                 fromalign=center,                 ngerman                 ]{scrlttr2} \usepackage{babel} \usepackage[utf8]{inputenx} \usepackage[right]{eurosym} \usepackage{tabularx} \usepackage{ltablex} \usepackage{booktabs} \usepackage{ifthen}  \input{Rechnungstabelle} \begin{document} \begin{letter}  \Rechnungstabelle{     1,0 h &amp; Do something &amp; \EUR{99,00} &amp; \EUR{99,00} \\      }     {99,00}     {}{} \end{letter} \end{document} Can I use Mage Hand to fly by Polymorphing myself and picking myself up with the hand? Could I first cast Mage Hand, then Polymorph into a Tiny creature less than 10 pounds then use the Mage Hand to pick myself up and fly at 30 Ft per turn?  Polymorph says:     The new form can be any beast ... The target's game statistics,   including mental ability scores, are replaced by the statistics of the   chosen beast. It retains its alignment and personality ... The   creature is limited in the actions it can perform by the nature of its   new form and it can't speak, cast spells, or take any other action   that requires hands or speech.   This keeps me from casting spells, which keeps me from just casting Mage Hand to fly as long as I want to over and over. But for that initial duration of the Mage Hand cantrip, could I use the Mage Hand to fly as a Tiny polymorphed creature until the spell runs out? Why does polynomial factorization generalize to matrices I'm reading about linear algebra and I came across with the following theorem where I have a problem convincing myself:      Theorem 2.1 $\,$ Every linear operator on a finite-dimensional complex vector space has an eigenvalue.      Proof:       To show that $T$ (our linear operator on $V$) has an   eigenvalue, fix any nonzero vector $v \in V$. The vectors $v, Tv,  T^2v,..., T^nv$ cannot be linearly independent, because $V$ has   dimension $n$ and we have $n + 1$ vectors. Thus there exist complex   numbers $a_0,...,a_n$, not all $0$, such that      $$a_0v + a_1Tv + ··· + a_nT^nv = 0.$$ Make the $a$’s the coefficients   of a polynomial, which can be written in factored form as $$a_0 + a_1z  + ··· + a_nz^n = c(z − r_1)\cdots(z − r_m),$$ where $c$ is a non-zero complex number, each $r_j$ is complex, and the equation holds for all   complex $z$. We then have      $${\color{red}{ 0=(a_0I + a_1T + ··· + a_nT^n)v= c(T − r_1I)\cdots(T −  r_mI)v}},$$ which means that $T − r_j$ I is not injective for at least   one $j$. In other words, $T$ has an eigenvalue.$\;\blacksquare$   I'm having trouble with the factorized form of the matrix polynomial (in red). I understood that the factorization holds for a polynomial by the fundamental theorem of algebra but why does this also hold for matrices?  In other words, why is the the part I highlighted true? Does such an factorization always exist?    Could I have some help to see this? Thank you =)   P.S.  is my reference (page 3).  UPDATE:   Someone else also has asked the  before it seems. Changing unusual foreign currencies in London I've just returned from several unusual countries in Asia, and annoyingly despite being told I'd have no problem, I'm unable to find anyone in London to change back to pounds.  I've tried Travelex in Heathrow, a currency exchange place, and the post office.  The currencies include cash from: Mongolia, Kyrgyzstan, Kazakhstan and Uzbekistan.  Most of the responses when shown the money is to the tone of "What the hell currency is this?" which doesn't fill me with confidence.  Any suggestions welcomed for places that may change this. Otherwise I may need to go back next year ;) Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}, \ \ a \in \mathbb{Q}$, which I don't think is possible, but how to show it? Determine resultant velocity of an elastic particle-particle collision in 3d space So I have two particles that have collided in 3 dimensional space. I want the particles to rebound off of each other in an elastic manner. How do I determine the resultant velocity vector if I know: Both particles' initial velocity vector, their masses, and their initial position vectors relative to the origin.  I tried to follow the instructions on  but I am not sure how to make their equations work without polar coordinates / how to translate my data to polar coordinates.  Also, if you answer my question in terms of polar coordinates, can you explain which angle is theta and which is phi? Since there are 2 different conventions, I am not really sure which angle is being referred to in the explanation on the aforementioned website.  EDIT: This isn't for a class. I'm writing a physics engine for a gravity simulation I am making using Unity. I haven't taken kinematics since I was in high school, which is why I am so rusty. Why do some statistics symbols have a "squared", e.g. Variance $\sigma^2$, "R squared" $R^2$ or heritability $H^2$ I can't get grub menu to show up during boot After trying (and failing) to install better ATI drivers in 11.10, I've somehow lost my grub menu at boot time.  The screen does change to the familiar purple colour, but instead of a list of boot options it's just blank solid colour, and then disappears quickly and boots into the default entry normally.    How can I get the bootloader back?  I've tried sudo update-grub and also various different combinations of resolutions and colour depths in startupmanager application with no success (640x480, 1024x768, 1600x1200, 16 bits, 8 bits, 10 second delay, 7 second delay, 2 second delay...)   edit:   I have already tried holding down Shift during bootup and it does not seem to change the behaviour.  I get the message "GRUB Loading" in the terminal, but then the place where the grub menu normally appears I get a solid blank magenta screen for a while.  Here are the contents of /etc/default/grub  # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 GRUB_HIDDEN_TIMEOUT=0 GRUB_HIDDEN_TIMEOUT_QUIET=true GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX=" vga=798 splash"  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM="0x01234567,0xfefefefe,0x89abcdef,0xefefefef"  # Uncomment to disable graphical terminal (grub-pc only) #GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass "root=UUID=xxx" parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY="true"  # Uncomment to get a beep at grub start #GRUB_INIT_TUNE="480 440 1"