Yes/No: Is $\mathbb{Q}[y]$ is field? We know that $\mathbb{Q} [x,y]/(x)$ is  isomorphic to $\mathbb{Q}[y]$.  My question is  that       Is $\mathbb{Q}[y]$ is field  ?  Yes/No   My attempt: I think  yes, because  we know  $ \mathbb{Q}[x] $ is a field and an integral domain  in a similar way  we can also said that $\mathbb{Q}[y]$ is a field and an integral domain  Is its true ?
Why can't the Polynomial Ring be a Field? I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This implies an inverse multiple for every Element in the Set.   The book doesn't elaborate on this, however. I don't understand why a Polynomial Ring couldn't have an inverse multiplicative for every element (at least in the Whole numbers, and it's already given that it has a neutral element). Could somebody please explain why this can't be so?
What is the inverse function of $e^x +x$? I have installed GCC compiler by installing the build-essential package.  After the installation I wrote a simple C program. I tried to run it with the following command:  gcc First.c ./a.out   but I'm getting a bash: ./a.out: Permission denied message. I don't know what to do now. Many-worlds: how often is the split how many are the universes? (And how do you model this mathematically.) How to show that there is an element of order $51$ in the multiplicative group $(Z/103Z)^∗.$  I can't use Cauchy's theorem since $51$ is not prime. Normalizing constant in Bayes theorem I read that in Bayes rule, the denominator $\Pr(\textrm{data})$ of   $$\Pr(\text{parameters} \mid \text{data}) = \frac{\Pr(\textrm{data} \mid \textrm{parameters}) \Pr(\text{parameters})}{\Pr(\text{data})}$$   is called a normalizing constant. What exactly is it? What is its purpose? Why does it look like $\Pr(data)$? Why doesn't it depend on the parameters? Towards the end of , how do I get  $$ \frac{\sqrt{8z + 1} - 1}{2} &lt; w + 1 $$  Here $w = x + y \geq 0, t = (w^2 + w)/2 , z = t + w, x,y \in \mathbb N_{\geq 0}$.  Thanks. Union of a finite set and a countably infinite set is countably infinite Ok, here is the problem statement:     Prove that if $S$ is any finite set of real numbers, then the union of $S$ and the integers is countably infinite.   This seems pretty obvious to me, knowing that 2 countable sets are countable.  But is there some step by step way to prove this?  Like do I need to prove bijectivity or something?  Thanks! Why is the derivative of a vector orthogonal to the vector itself? $R(t) \cdot R'(t) = 0$, which is what every source I can find tells me. Even though I understand the proof I don't understand the underlying concept. If $R(t)\cdot R'(t) = 0$, then $R'(t)$ is orthogonal to $R(t)$, right?   But you use the same derivative to find the tangent of a curve. Then somehow if you differentiate the tangent itself, you get the normal to the curve.  I really can't wrap my head around this. Could someone help me understand? Browsing folders with more than 100000 images I would like to do two things   Increase the size of inmargin area (i.e., customize inner sidebar so that more contents could fit there) Reduce the empty space between the title of block environment and the slide edge (from the example below, the title of block has too much of wasted space to its left which could be used to create more readable sentences)     MWE  \documentclass[xcolor={x11names}]{beamer}  \usecolortheme{structure} \useinnertheme{rectangles} \useinnertheme{inmargin}  \usecolortheme{crane}  \usefonttheme{structurebold}   \usepackage{fontspec}  \usepackage{calc}  \usepackage[inline]{enumitem} % Description list defaults      % To change font     \newcommand{\descriptionfont}{\fontencoding{T1} \fontfamily{bch} \selectfont}      % Label indentation     \newcommand{\descLabOffset}{\widthof{\hspace{2.6mm}}}      % Left of description aligned with the left margin     \newcommand{\descLeftLabMarInd}{0mm - \descLabOffset}      % Left of description aligned with \parindent     \newcommand{\descLeftLabParInd}{\parindent - \descLabOffset}      % Set the list options     \setlist[description, 1]{font=\descriptionfont, leftmargin=*, labelsep=3mm, labelindent=\descLeftLabMarInd, align=left}  \begin{document}   \begin{frame}{Induction Motors}      \begin{block}{Why is induction motor very common?}          Induction motors are very practical for the following reasons          \begin{description}             \item[Rigid] Rigid             \item[Cheap] Cheap             \item[Low Maintenance] Low Maintenance             \item[Self-Starting] Self-starting             \item[No Excitation Needed] No Excitation Needed         \end{description}      \end{block}  \end{frame}  \end{document} $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$ What does the human body use oxygen for besides the final electron acceptor in the electron transport chain? My biology teachers never explained why animals need to breathe oxygen, just that we organisms die if we don't get oxygen for too long. Maybe one of them happened to mention that its used to make ATP. Now in my AP Biology class we finally learned the specifics of how oxygen is used in the  due to its high electronegativity. But I assume this probably isn't the only reason we need oxygen.   What other purposes does the oxygen we take in through respiration serve? Does oxygen deprivation result in death just due to the halting of ATP production, or is there some other reason as well? What percentage of the oxygen we take in through respiration is expelled later through the breath as carbon dioxide? The code below compiles, but has different behavior for the char type than for the int types.  In particular     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;   result in 3 instantiations of templates for three types: int8, uint8 and char. What gives?  The same is not true for ints: int and uint32 which result in the same template instantiation, and signed int another.  The reason seems to be that C++ sees char, signed char and unsigned char as three different types. Whereas int is the same as a signed int. Is this right or am I missing something?  #include &lt;iostream&gt;  using namespace std;  typedef   signed char       int8; typedef unsigned char      uint8; typedef   signed short      int16; typedef unsigned short     uint16; typedef   signed int        int32; typedef unsigned int       uint32; typedef   signed long long  int64; typedef unsigned long long uint64;  struct TrueType {}; struct FalseType {};  template &lt;typename T&gt; struct isX {    typedef typename T::ikIsX ikIsX; };   // This  int==int32 is ambiguous //template &lt;&gt;            struct isX&lt;int  &gt;    { typedef FalseType ikIsX; };  // Fails template &lt;&gt;            struct isX&lt;int32  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint32 &gt;  { typedef FalseType ikIsX; };   // Whay isn't this ambiguous? char==int8 template &lt;&gt;            struct isX&lt;char  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;int8  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint8 &gt;  { typedef FalseType ikIsX; };   template &lt;typename T&gt; bool getIsTrue(); template &lt;&gt;           bool getIsTrue&lt;TrueType&gt;() { return true; } template &lt;&gt;           bool getIsTrue&lt;FalseType&gt;() { return false; }  int main(int, char **t ) {    cout &lt;&lt; sizeof(int8) &lt;&lt; endl;  // 1    cout &lt;&lt; sizeof(uint8) &lt;&lt; endl; // 1    cout &lt;&lt; sizeof(char) &lt;&lt; endl;  // 1     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;     cout &lt;&lt; getIsTrue&lt; isX&lt;int32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;int&gt;::ikIsX  &gt;() &lt;&lt; endl;  }   I'm using g++ 4.something Version of the app: 1.0.51. (It is up to date at the time of writing according to Google Play.)  This started happening after the network problem that happened earlier this morning. Both my tablet and my phone are popping up notifications for events (comments, and probably edits too) that happened prior to the network issue. I had received notifications for these events already and had already cleared them before the network issue happened.  The repeated notifications do not show up in the bar displayed at the top of the Stack Exchange sites. In other words, I get notified again in Android but do not get notified again on the SE web sites. Getting "'file' is different from the same file on the system" when I attempt to update through synaptic Encountered an error while updating through Synaptic Manager. Subsequent updates gave the error at the bottom of the output. I'm a noob so any help is appreciated.  al@al:~$ sudo apt-get -f upgrade Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following NEW packages will be installed:   libjack-jackd2-0:i386 The following packages have been kept back:   linux-headers-generic skype 0 upgraded, 1 newly installed, 0 to remove and 2 not upgraded. Need to get 0 B/202 kB of archives. After this operation, 493 kB of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 310056 files and directories currently installed.) Unpacking libjack-jackd2-0:i386 (from .../libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb) ... dpkg: error processing /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb (--unpack):  './usr/share/doc/libjack-jackd2-0/buildinfo.gz' is different from the same file on the system dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing:  /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1) The reference to entity "foo" must end with the ';' delimiter Trying to format USB (using Linux/fdisk) so it shows up on my Mac I have a USB drive that I can see/access/modify/etc in Linux/Ubuntu but it doesn't show up on my Mac OSX.  I did sudo fdisk /dev/sdb (where my USB is) and then g to create a new GPT partition table, n to make a new partition, used all default settings, then w to save and exit.  Then I did sudo mkfs.fat -F32 /dev/sdb1 to format the partition with a FAT32 filesystem.  I ejected the USB and put it in my Mac and it doesn't show up / get detected.  What did I miss? Why are all of my appliances becoming energized when ground wire is excluded? Add a section before \tableofcontents to content Can you find all classes in a package using reflection? Is it possible to find all classes or interfaces in a given package? (Quickly looking at e.g. , it would seem like no.)