Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be $\mathbb{Q}(\sqrt5)$ obtained by the relation between the golden ratio and $\cos(2\pi/5)$ (the minimal polynomial of the golden ratio $\phi$ is $x^2-x-1$ and so the extension is quadratic).   Also $\cos(\pi/5)=\phi/2\implies \cos(2\pi/5)=\phi^2/2-1$ by double angle identity.      Is it true that in general the quadratic extension is $\mathbb{Q}(\sqrt{\pm p})$ where $\pm$ depends on whether $p$ is $1$ or $-1$ $\pmod 4$?    Please prove or disprove.
I want to prove the assertion:      The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.   My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and $$ p = \epsilon^{-1} (1-\zeta_p)^{p-1} $$ where $\epsilon$ is a unit.  Since $p$ is an odd prime, $(p-1)/2$ is an integer and  $$ \sqrt{\epsilon p } = (1-\zeta_p)^{(p-1)/2} $$ makes sense and belongs to $\mathbb{Z}[\zeta_p]$.    How do I deal with the $\epsilon$ under the square root?  I guess the condition on the congruence class of $p$ comes from that.  Is this even the right way to proceed?  Uniqueness is not clear to me either.  I thought about looking at the valuation $v_p$ on $\mathbb{Q}$, extending it to two possible quadratic extensions beneath $\mathbb{Q}(\zeta_p)$, then seeing how those have to extend to common valuations on $\mathbb{Q}(\zeta_p)$, but I didn't see how to make it work.  I would appreciate some help.
Execute a Java program from our Java program I used   Runtime.getRuntime().exec("_____")   but it throws a IOException as below:  java.io.IOException: CreateProcess: c:/ error=5   at java.lang.Win32Process.create(Native Method)   at java.lang.Win32Process.&lt;init&gt;(Win32Process.java:63)   at java.lang.Runtime.execInternal(Native Method   I don't know whether I have the problem with specifying the path or something else. Can anyone please help me with the code. Open subsets in a manifold as submanifold of the same dimension? An open set in an $n$-manifold is clearly a submanifold of the same dimension as its containing manifold (see open manifolds).   Now, given an $n$-manifold $M$, is it true that a set, to be the underlying set of a submanifold of $M$ with dimension $n$, must be open? True or false: {{∅}} ⊂ {∅,{∅}} Training involved to ride 175 miles (281 km) in a day I want to be able to ride 175 miles (281 km) in a day (to do the ). I used to be able to do about 60 miles (96.5 km), only once hitting 80 miles (128 km), but I'm out of shape now. Wondering how I can systematically train to get to 175 miles at an average of 15 mph (24 km/h) Are there guides out there to help with this?  What's some advice?  [update] The  will happen each year, so my intention isn't to ride this year, rather what do I need to do to ride it in some future year? Recovering *permanently* deleted emails from Gmail person in distress here.  Dont ask me how, but I ended up deleting all my 12,000 inbox emails from Gmail into Trash and then erased my Trash as well. Gmail gave me a warning box and took 20 seconds to delete. This was not what I intended. I have searched high and low to check if there is any way of recovering my 10 years worth of emails and could not find any.  Any one here know of a way, ANY WAY, to recover my accidental permanent deletion of all my 10 years worth of email (deleted from Trash as well)? How to block specific notifications from an app? A certain app I use gives me crucial notifications that I need, but also notifies me almost daily that over 80% of my RAM is used. I would like to allow this app to send me all notifications except those that contain the string "80% of your RAM" in the text of the notification.  Obviously, there is no way to filter a certain app's notifications in stock Android (other than blocking all of a certain app's notifications, but that's not what I want!), but is there a way that I can block notifications if they contain a certain string of text?  Nexus 6P running Android Marshmallow 6.0.1, if you're curious the app in question is . How can I re-write the conditionals for this button I've been trying out React for a couple of days and I tried to build a button component with some conditionals. Everything works but I feel like I am not doing it the right way. Could anyone review my code and give me some advice and tell me what I am doing wrong or what i could do better. I think I am using too many return statements. I use Typescript for type checking. Would appreciate any help! import { Link } from &quot;gatsby&quot;;  interface Props {   content: string,   color: string,   path?: string,   type?: string }  export default function Button(props: Props): JSX.Element {   let elementType;    const checkProps = () =&gt; {     const availableColors = ['blue', 'grey', 'dark-grey', 'white'];     const borderColors = [''];     let buttonStyle;      // Check if props.color exists in availableColors     if(!availableColors.includes(props.color)) {       console.error(`&quot;${props.color} color key not available for button style`);       return;     }          buttonStyle = buttonStyles[`button--${props.color}`];      // Check if props.content is valid     if(!props.content) {       console.error(`&quot;${props.content} is not valid as content`);       return;     }      // Check if props.type exists     if(!props.type) {       elementType = &lt;button className={ buttonStyle }&gt;{ props.content }&lt;/button&gt;       return;     }      if(props.type !== 'link') {       console.error(`&quot;${props.type} is not a valid type, use 'link' instead or use nothing for default button`);       return;     } else {         elementType = &lt;Link to={ props.path } className={ buttonStyle }&gt;{ props.content }&lt;/Link&gt;     }   }    checkProps();    return (     &lt;React.Fragment&gt;       { elementType }     &lt;/React.Fragment&gt;   ); }``` How to boot WUBI installed Ubuntu within Windows 8? I have Windows 8 Release Customer Preview on my laptop. I tried to install Ubuntu on it with Ubuntu Windows installer (I have always used this to install, although that was on Windows 7), it installs and asks to reboot, then immediately Windows 8 boots up.  I do not have the option to choose what OS to boot. How can I boot to Ubuntu? VM is not exactly what I would like to use, but it is an option, however a last resort. Any help? Ratio test and the Root test I once learnt, that it is uncool to use $$, but why is that? Why does \[ not act as a abbreviation to \begin{align}? I noticed, that there is a difference, since one cannot use &amp; and \\ inside a block started with \[. Is Dishonored an open-world game? I was considering whether or not to buy Dishonored, and watching gameplay of it on Youtube, when a thought popped into my head: is Dishonored free-roam? Can you go anywhere you want, similarly to Fallout or The Elder Scrolls? Implementing i18n string translation in a custom module? I have a custom module I've built which essentially acts as a repository of custom CTools keywords that users can arbitrarily define. The issue I'm facing is I need to be able to translate these user-defined strings, so I think I need to implement string translation using the i18n_string module.  I have never worked with translation or localization before, so I'm a bit lost with regards to going about how to implement this. I think I'm on the right path with i18n_string though. Can someone experienced with i18n provide examples that I can look at so I can learn the proper way to implement this? Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field How can I find just the *.mp3 files with a recursive command like ls -R *.mp3 in a directory that contains several sub-directories and at the end copy these files in a directory that I choose.  Thanks for your support. How do I install Ubuntu Server (step-by-step)? I'd like to install Ubuntu server on my machine, is there a step-by-step instructions on how to accomplish that or a guide that will just go through the basic steps of installing Ubuntu Server from beginning to end.  Instructions originally for 13.04. Do we use the SD of whole population or SD of just one sample to calculate SE of samples means in central limit theorem? I'm learning applications on Central Limit Theorem and got really confused with a few points. According to this , the procedure to apply CLT usually goes like this:   So if SD is the population standard deviation, how are we gonna get it?? Isn't the whole population standard deviation what we eventually calculate by applying CLT and analyzing a sample of the whole population? How come the population standard deviation become a prerequisite??   Please tell me this tutorial is wrong.   I think the SD actually refers to the standard deviation of a sample (of some size n), which we can actually get easily. For example:    sample the whole population with a sample size of n (e.g. randomly select 10000 users from the whole population of 10 billions) calculate the mean of the 10000 measurements. calculate SE = SD / sqrt(10000), where SD is the standard deviation of the 10000 measurements, instead of the standard deviation of the whole population of 10 billions.    This explanation would make much more sense.   Any thoughts? Generate a comma-separated list of cell contents, excluding blanks To concatenate a certain four cells, I'd use:  =CONCATENATE(A2,",",C2",",D2,",",F2)   This would make it so that...   A2 = "Matthew" C2 = "Mark" D2 = "Luke" F2 = "John"   would result in Matthew,Mark,Luke,John.  But we run into problems with something like...   A2 = "Jesus" C2 = "" D2 = "Mary" F2 = "Joseph"   which would result in Jesus,,Mary,Joseph.  Here, the extra comma is undesired. Is there a way to gracefully handle this so that all non-blank cells are included in the comma-separated list, while avoiding the addition of unnecessary commas when some cells are blank?  Certainly, this can be done with a fair number of nested IFs, but I really want to avoid that if possible. Can it be done with native Excel functions, or perhaps an array formula? Or would one have to resort to VB Script for something like this? How to use Alt+Shift to switch keyboard layouts? I recently upgraded to Ubuntu 13.10, with the new keyboard handling system. The little "switch keyboard" option does not let me set it to Alt+Shift. How can I get this keyboard binding, as I have absolutely no interest in using the Mac-esque Super+Space combo? Is there a problem with closing the lid of a MacBook Pro when using an external monitor? I am using an external monitor and the only way (that doesn't involve jumping through hoops) to turn off the MacBook Pro monitor is to close the lid.  Does closing the lid causes any problems? I have read the following on :     You can still run a single external monitor if you close the lid of   your laptop, but that restricts airflow and exposes your display to a   lot of heat. Not good. How do I find out which DOM element has the focus?