I would like some feedback on my proof of the fact stated in the question.  First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know that $ \mathrm{Gal}(\mathbb{Q}(\zeta_5)) \cong C_{4} $. By the fundamental theorem of Galois theory, the subfields of $ L $ of degree 2 are in one-to-one correspondence with the subgroups of the Galois group which have $ 4/2 = 2 $ elements. However, $ C_4 $ has only one subgroup of order 2, which is $ \{e, g^2\} $ where $ g $ is an arbitrary generator. We conclude that $ L $ has only one subfield of degree 2, and it is fixed by the aforementioned subgroup of the Galois group.  Now, the Galois group is comprised of automorphisms $\sigma_i: \zeta_5 \to \zeta_5^i $ for $1 \leq i \leq 4 $, and this group has only one element of order 2, which is $\sigma_4$. We conclude that the intermediate field $ F $ is the one fixed by the cyclic subgroup generated by $ \sigma_4 $. Now, we consider the polynomial $ P(X) = (X - \zeta_5)(X - \zeta_5^4) $. Again, by the fundamental theorem, $\textrm{Gal}(L/F) = \{\sigma_1, \sigma_4\}$. We see that the entire Galois group fixes $P$, so $P$ has coefficients in $F$. Defining $ A = \zeta_5 + \zeta_5^4 $ which is negative of the coefficient of $X$ in $P$, we see that  $ A^2 + A - 1 = \zeta_5^2 + \zeta_5^3 + \zeta_5 + \zeta_5^4 + 1 = 0 $  and therefore, by the quadratic formula, $$ A = \frac{-1 \pm \sqrt{5}}{2} $$ and $ \sqrt{5} \in F $, implying that $ \mathbb{Q}(\sqrt{5}) = F $ as $ F $ has degree 2.
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.
Get referenced node fields in twig In Drupal 8, I have a content type, that has an "entity_reference" field and I wish to access the referenced node's fields in twig template. the kint() output for the field is:   In this list I can't find the referenced node's field values. How can I (if at all possible) load the node or node fields/values in twig?  Best,  Alari I have 3 accounts that I am deleting - however, several hours after the 24 hour (+ grace period - end about 4-5 hours ago), the countdown completed, but disappeared and the accounts are still there.   I would rather not go through another 24 hours, and would like to have these accounts deleted (and yes, I have thought about it and this decision is not up for discussion).  Currently, the accounts are still there.  Could this please be seen to? How can I recover a game center account where I forgot ID and password? A long time ago, I made a game center account to which I attached my CoC village. This account hasn't been used anymore, and I have forgotten the ID and the password.  Now I want to transfer that game to my new phone which has a new game center ID, but need a transfer code from the old account to do so. Is there a way to recover my old game center account details? Do I need a UK or Ireland transit visa when flying from the US to a Schengen country? I am travelling to Budapest and therefore have a Schengen Visa. I hold an Indian passport and a valid US visa. I'm travelling from Chicago-Heathrow-Budapest and returning by Budapest-Dublin-Chicago.  Would I need a transit visa for UK/Ireland?  UPDATE  Hoping this will help someone someday.   I was not allowed to board my return flight (BUD-DUB-ORD) because I do not hold an overnight Irish Transit Visa. They said I could have transited happily if the connecting flight was in the same day, but because I would have to originally land at 10 PM and catch the connecting flight next morning at 10 AM, I needed a transit visa. Well, I thought it was BS, but couldn't help comply with the rules. I booked another last minute ticket and flew via Warsaw via LOT. Meta-analysis of means and medians in R? I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.   Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.   I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?  Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians. Which Harry Potter works are considered canon? There are many different Harry Potter works including:   books short stories movies card games video games toys amusement parks web content   Which ones are considered to be canon? Maximum power transfer and Thevenin Simulating draws from a Uniform Distribution using draws from a Normal Distribution I recently purchased a  in which one of the probability questions was as follows:     Given draws from a normal distribution with known parameters, how can you simulate draws from a uniform distribution?   My original thought process was that, for a discrete random variable, we could break the normal distribution into K unique subsections where each subsection has an equal area under the normal curve. Then we could determine which of K values the variable takes by recognizing which area of the normal curve the variable ends up falling into.   But this would only work for discrete random variables. I did some research into how we might do the same for continuous random variables, but unfortunately I could only find techniques like inverse transform sampling that would use as input a uniform random variable, and could output random variables from some other distribution. I was thinking that perhaps we could do this process in reverse to get uniform random variables?  I also thought about possibly using the Normal random variables as inputs into a linear congruential generator, but I'm not sure if this would work.  Any thoughts on how I might approach this question? How to extend USB beyond the recommended maximum cable length of 5m? I'm planning to use USB port for an application(from a device to a PC) but the distance between the device and the PC will be around 15m. The recommended maximum USB cable length is 5m.  How can I buffer the data without loosing any information, safely? Would a USB hub work? Are there any disadvantages to always using nvarchar(MAX)? In SQL Server 2005, are there any disadvantages to making all character fields nvarchar(MAX) rather than specifying a length explicitly, e.g. nvarchar(255)? (Apart from the obvious one that you aren't able to limit the field length at the database level) For some reason my Steam just broke today after working perfectly since I installed it a few days ago.  It was originally installed via the deb from the Steam site. So I uninstalled that using the Ubuntu Software Center. Then I installed steam-launcher. When I run it from the menu it does nothing at all. When I go through terminal it gives this message, no error:  $ steam/usr/bin/steam:  line 196: /home/andrew/.local/share/Steam/steam.sh: Success   So I purge-removed it and removed the .steams directory from my home directory but on re-installation its the same story.  I am using Xubuntu 12.10 64bit.  It appears that steam .local/share/Steam/steam.sh is blank. What is it meant to have in it? I am so confused with how reinstalling it doesn't fix it. Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$ For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of  $p$   So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to show that.Can anybody help me this method?Any other type of prove is also welcomed. Wayne Dalton Garage So I recent had a Garage door "professional" come out to my house and look at my garage. I originally thought I was having a motor problem because it would only lift door sporadically. After performing a 25 point inspection he believed he located the problem, which was the spring system from my Wayne Dalton Garage (torquemaster). He told me how horrible of a system they are and they don't fix them because of that they only replace the entire system and would cost me $700 or more. So now I'm left trying to figure out what I need, torquemaster springs, bracket assembly, both or if I need something completely else. I need some experience Wayne Dalton experience. Every time I render an image and try to save it, it just disappears. How do you exactly save a render? I am using a mac. What's the full-form & significance of USS & ISS prefixes in name of Star Trek starships? In the Star Trek normal universe, almost all of Starfleet's starships were named with the prefix USS: Count from . Some popular ones: USS Enterprise - series, USS Stargazer, USS Exeter, USS Defiant, USS Excelsior, USS Voyager, USS Titan, USS Reliant, USS Equinox, USS Grissom, USS Relativity, etc.  In the mirror universe, Starfleet's starships were named with the prefix ISS. We don't have a big list, but we do see these: ISS Enterprise NX-01, ISS Enterprise NCC-1701, ISS Avenger.  What's the full form of USS &amp; ISS in Star Trek canon?  In the real world, the US Navy uses USS in ship names, which is reasonable as "United States Ship", but why is it used in Star Trek canon (in which there's no US to control starships)?  In the mirror universe, USS is replaced with ISS. Why? Does ISS have a meaning opposite from that of USS? Why is Newton's first law necessary? Newton's second law says $F=ma$. Now if we put $F=0$ we get $a=0$ which is Newton's first law. So why do we need Newton's first law ?   Before asking I did some searching and I got this: Newtons first law is necessary to define inertial reference frame on which the second law can be applied.  But why can't we just use Newton's second law to define an inertial frame? So if $F=0$ but $a$ is not equal to 0 (or vice versa), the frame is non-inertial.  One can say (can one?) we cannot apply the second law to define a reference frame because it is only applicable to inertial frames. Thus unless we know in advance that a frame is inertial, we can't apply the second law.  But then why this is not the problem for the first law?  We don't need to know it in advance about the frame of reference to apply the first law. Because we take the first law as definition of an inertial reference frame.  Similarly if we take the second law as the definition of an inertial frame, it should not be necessary to know whether the frame is inertial or not to apply the second law (to check that the frame is inertial). How does the simulation hypothesis relate to the ethics of simulated violence? Say, for the sake of this question,  is real. Are we then hurting "persons"/conscious beings in "our simulations" - video games - when we fight them? Why (not)? If so, is this unethical? Think for example about shooting an alien in the game No Man's Sky.  Edit/answer on the comment of Joseph Weissman: My reasoning is like this:   It is unethical to harm a person (with a person being everything that has ). By the simulation hypothesis (assuming it's true), everything around us (everything that we know of), is simulated. By 1 and 2, all persons we know of are simulated. So it's unethical to harm the simulated persons around us. A game is a simulation just like we are in a simulation, so the game has simulated persons. Therefore -  - it's unethical to harm the persons in a video game.   I know that the premises - besides the simulation hypothesis itself - are debatable, but that's the reason I asked the question in the first place.  I was also thinking that the characters in the video game differ because they have far less to no intelligence, that much of it is scripted. But I neither have a proof that we ourselves aren't scripted, with free will being an illusion. Also, how intelligent does a character in a simulation/video game has to be to have personhood then? I think this is a case of the .  Just to be sure, please note that this question is not about whether the simulation hypothesis is true or how likely it is to be true. It is about (one of the) consequences if the simulation hypothesis is true. It's also not a question about violence in games in general. I am trying to remove all files and subdirectories in a directory. I used rm -r to remove all files, but I want to remove all files and subdirectories, excluding the top directory itself.  For example, I have a top directory like images. It contains the files header.png, footer.png and a subdirectory.  Now I want to delete header.png, footer.png and the subdirectory, but not images.  How can I do this in linux? Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism. Intuition on proof of Cauchy Schwarz inequality To prove Cauchy Schwarz inequality for two vectors $x$ and $y$ we take the inner product of $w$ and $w$ where $w=y-kx$ where $k=\frac{(x,y)}{|x|^2}$ ($(x,y)$ is the inner product of $x$ and $y$) and use the fact that $(w,w) \ge0$ . I want to know the intuition behind this selection. I know that if we assume this we will be able to prove the theorem, but the intuition is not clear to me.