I need some help with starting this problem. I am given that $$C=\left\{\left( \begin{array}{cc} a &amp; b \\ -b &amp; a \\ \end{array} \right): a,b\in\mathbb{R}\right\}$$ is a subring of the ring of all $2\times2$ matrices with real number entries, with respect to usual addition and multiplation of matrices. My task is to show that $C$ is isomorphic to the field $\mathbb{C}$ of complex numbers.
Below is an example from I.N. Herstein:       Let  $R=\Bigg\{\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)\Bigg|a,b\in \mathbb R\Bigg\}$ and let $\mathbb C$ be the field of complex numbers. Define $\psi :R\to \mathbb C$ by $\psi\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)=a+bi.$       It is asked to prove that $\psi$ is a ring isomorphism of $R$ onto $\mathbb C$ .    can anyone help me with some hint how to prove the ring isomorphism.
Having trouble finding Plantera's Bulb I have been playing on hardmode for a while and have beaten the three  a couple of times. But I have yet to find a . It's getting really frustrating.  Does anyone have any methods to finding a Plantera Bulb? A Polygon is inscribed in a circle $\Gamma$ A regular polygon P is inscribed in a circle $\Gamma$. Let A, B, and C, be three consecutive vertices on the polygon P, and let M be a point on the arc AC of $\Gamma$ that does not contain B. Prove that  $MA\cdot MC=MB^2-AB^2$  I inscribed the polygon P in the unit circle and let B=1. Past this, I'm not really sure how to proceed. It might be helpful to have M at 1 as well but given that the polygon would have the n-th roots of unity in it B seemed like a good choice. What would be a good way to proceed from here? Can there be true conclusions without assumptions? I was thinking of the sentence  "I think therefore I am",  which I had for a long time considered indisputable because it's self-evident.   Then I considered the hypothetical situation where my thoughts are illogical – and of course I can't be aware that they are – and so to a logical outsider I am not saying something that is actually indisputable and self-evident, but in fact may be total nonsense. Perhaps there is a hypothetical evil demon altering my ability to reason. However convincing it seems to me that I MUST exist since, in whatever since "I" or "exist" is, there is the ability to question whether I do, the conclusion can't actually be trusted without doubt since I cannot know if I am in fact reasoning logically.  Even what I consider to be trivial tautologies require this assumption of sanity, and therefore there is no true conclusion without assumption – an assumption about the nature of my own reasoning. The only absolute truth I have is unspeakable subjective experience, and any articulation (and I realize the apparent circular reasoning I've cornered myself into here...) is necessarily uncertain.  I realize that perhaps this doesn't even make sense, but was just curious if anyone had entertained this line of thought and come to some satisfying resolution. A potentially dangerous Request.Form value was detected from the client Every time a user posts something containing &lt; or &gt; in a page in my web application, I get this exception thrown.  I don't want to go into the discussion about the smartness of throwing an exception or crashing an entire web application because somebody entered a character in a text box, but I am looking for an elegant way to handle this.  Trapping the exception and showing     An error has occurred please go back and re-type your entire form again, but this time please do not use &lt;   doesn't seem professional enough to me.  Disabling post validation (validateRequest="false") will definitely avoid this error, but it will leave the page vulnerable to a number of attacks.  Ideally: When a post back occurs containing HTML restricted characters, that posted value in the Form collection will be automatically HTML encoded. So the .Text property of my text-box will be something &amp; lt; html &amp; gt;  Is there a way I can do this from a handler? Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from. How do I get a tourist visa (or a tourist card) for Cuba?  I live in the Netherlands, but I'm not a Dutch citizen, I have an Italian passport. Going back twice required to return to search results in the tags view How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected?    Note that $X$ is a topological space with subspace topology $\tau = \left\{ U \cap X \mid U\text{ is open in }\mathbb{R}^2\right\}$.  I only know two things (with respect to path connectedness):   A topological space $X$ is path connected iff for every $p,q \in X$, there is a continuous function $f: [a,b] \to X$ with $f(a)=p$ and $f(b) = q$.    Continuity $f$ is defined as for every open set in $X$, its preimage is open in $[a,b]$ (with subspace topology induced by $\mathbb{R}$ (with standard topology) )   Suppose $X$ and $Y$ are two topological spaces and $f: X \to Y$ is a continuous surjection. If $X$ is path connected, then $Y$ is path connected. When trying to launch Audacity, I get the following error:  Audacity could not find a place to store temporary files. Please enter an appropriate directory in the preferences dialog.   No files or folder have been moved.   I have purged Audacity, deleted any files, etc manually and re-installed   In terminal:  pst007x@pst007x-Serval-Professional:~$ audacity   (process:16390): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::sm-connect after class was initialised  (process:16390): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::show-crash-dialog after class was initialised  (process:16390): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::display after class was initialised  (process:16390): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::default-icon after class was initialised  ** (Audacity:16390): WARNING **: Owner of /tmp/orbit-pst007x is not the current user  Segmentation fault pst007x@pst007x-Serval-Professional:~$    I have not changed any users or added any additional users. All privaleges, etc are default.   Ubuntu 13.10 64bit (all updates applied)  ADDITIONAL:  I can open as gksu audacity  So it looks like privileges is an issue somewhere. I have checked all privileges and all seems ok, but clearly not....  I tried changing the Directory in Preferences to a directory in my home folder:  /home/pst007x/Music/audacity_temp/  but when starting again (not in root) same error...  Any ideas what files could be lacking privileges?   After changing the Directory, when running I get this:  pst007x@pst007x-Serval-Professional:~$ audacity   (process:28364): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::sm-connect after class was initialised  (process:28364): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::show-crash-dialog after class was initialised  (process:28364): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::display after class was initialised  (process:28364): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::default-icon after class was initialised Segmentation fault pst007x@pst007x-Serval-Professional:~$    I checked installed files using Synaptic Package Manager, then checked permissions:  pst007x@pst007x-Serval-Professional:~$ sudo chmod -R 777 /usr [sudo] password for pst007x:  pst007x@pst007x-Serval-Professional:~$ audacity  (process:31026): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::sm-connect after class was initialised  (process:31026): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::show-crash-dialog after class was initialised  (process:31026): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::display after class was initialised  (process:31026): GLib-GObject-WARNING **: Attempt to add property GnomeProgram::default-icon after class was initialised Segmentation fault pst007x@pst007x-Serval-Professional:~$    EDIT. After the above, I rebooted and was no longer able to login. So I did a fresh install. Not a bad thing since the original install on the this system was 10.10. Working ok after fresh install.  Thank you for taking the time to post answers and suggestions.. How to access a clicked URL in a URL handler application created in Automator I'm currently trying to modify the default URL handling behavior in OS X. I would like, when clicking on any link outside of a web browser, to copy this URL to the pasteboard instead of opening it in the default web browser – for instance, clicking a link in a Skype conversation should copy this URL to the OS X pasteboard instead of opening it in Firefox (which is currently my default web browser).  I have thought about solutions, but I’m currently unable to implement one. If I can create an application that puts URLs into the pasteboard that are sent to it, and can set this application as the default web browser, I would be able to achieve my goal.  I created an application with Automator that executes the shell command pbcopy and have set this application as my default web browser, but I didn’t get the expected result. I did some testing and I can confirm that the application runs on clicking on a link, however, I cannot get the URL passed to the application / shell script. I tried both stdin or a passed argument, but without success.  Is there a way for an application created with Automator to retrieve a clicked URL and send it to pbcopy, when that application is set as a (default) handler for URLs?  Alternatively, is there a better way to create a fake web browser that takes the input and puts it in the pasteboard than Automator? Or is there any other way that I can achieve my goal without a custom application / script? How to interpret a quantum circuit as a matrix? Sometimes I use, $PROJECT_HOME/* to delete all files in the project. When the environment variable, PROJECT_HOME is not set (because I did su and the new user doesn't have this environment variable set), it starts deleting all files from the root folder. This is apocalyptic.  How can I configure bash to throw error, when I use an undefined environment variable in the shell? How can I add an empty directory to a Git repository? How can I add an empty directory (that contains no files) to a Git repository? I have a statement,     Either p or q   and I have to write it in terms of logical connectives but I don't get which logical connector should I be using? Here is what I did (I think there could have been a better way to do this)     $(p \lor q ) \land (\neg((p \Rightarrow q) \land (q \Rightarrow p)))$ What happened to the Grid at the end of Tron Legacy? Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$ Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. Assume  $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times\ldots$$ $$G_2= \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \ldots$$ How do I prove $G_1$ and $G_2$ aren't isomorphic? I asked  this question here  and I received tree answer, but the answer of Ludolila will be complete when  $G_1$ and $G_2$ are not isomorphic.    Thanks in advance Why and how does C# allow accessing private variables outside the class itself when it's within the same containing class? Irreducible polynomial which is reducible modulo every prime How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$?   For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not solutions of $x^4+1=0$ then if it is reducible the factors are of degree $2$. This gives that $x^4+1=(x^2+ax+b)(x^2+cx+d)$ and solving this system of equations $\bmod 3$  gives that $x^4+1=(x^2+x+2) (x^2+2x+2) \pmod 3$. But is there a simpler method to factor $x^4+1$ modulo a prime $p$? Regular cast vs. static_cast vs. dynamic_cast I've been writing C and C++ code for almost twenty years, but there's one aspect of these languages that I've never really understood. I've obviously used regular casts i.e.  MyClass *m = (MyClass *)ptr;   all over the place, but there seem to be two other types of casts, and I don't know the difference. What's the difference between the following lines of code?  MyClass *m = (MyClass *)ptr; MyClass *m = static_cast&lt;MyClass *&gt;(ptr); MyClass *m = dynamic_cast&lt;MyClass *&gt;(ptr);