Using Merge to Combine Associations
How to organically merge nested associations?
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. If I am thinking about using one of the 'fancy' top level domains such as .me or .tv should I be worried about how Google treats such domains?  Or to put it another way, if all other things are equal (good content, links etc.), will Google regard a .tv or .me as equal to a traditional .com or co.uk site? How does the Mac choose which connection to use when both Wifi and Ethernet are connected? When I'm connected to two networks with separate routers using Wifi and Ethernet, how does my Mac know what connection to use when I visit a website? Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$ Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$    JUNK  Guessing asking for $8/9 * k = 0 $ (being a whole integer) for smallest integer so for this one is 9 . another guess would be the smallest element of $Q/Z$ being something like $9/8$  $14/5$ would be 5  $48/28=12/7$ would be 7? Agreement in "[Singular Noun] Is/Are [Plural Noun]"? My fish's native habitat is rice fields. My fish's native habitat are rice fields.   Which one is correct? I'm pretty sure it's the first, since 'is' modifies 'habitat,' but it still sounds weird... How do I customize the date/time display in GNOME Shell's top bar? Time indicator is shown as "%a %H:%M" in the center of the top bar.  How do I move it back to the right where it belongs, and make it include the full date and time ("%Y-%m-%d %s %H:%M")? table of contents no word separation Is it possible to have in the toc no word separation but have a word separation in the chaptername:  For example:   Contents  1. max Mustermann   Chapter  Max Muster- mann Radius of Convergence of a given Power series The number of nonnegative integer triples with sum equal to $12$ How many triples $x,y,z$ satisfy  $x+y+z=12$, where  $x,y,z\ge 0$ are integers? Note that these are not ordered: $ (12,0,0)$ and $(0,12,0)$ are treated as same. Progress I have tried the formula ${13\choose 2}$ to select values of $x$ and $y$ from $0$ to $12$ as  $z$ would keep changing according to $x$ and $y$. (That is, $z=12-x-y$.) The problem is that that certain combinations of $x$ and $y$ would give value of $x+y$ greater than $12$. I can't seem to get around this. Bash - Integer expression expected Are the Yongnuo clones of Canon flash equipment and lenses legal to sell/buy? On thanksgiving day a woman posted in a facebook group that her father had just passed away (left her money) and in good deed because it was thanksgiving she was offering to pay people’s debts. Well I inboxed her and she pretty much wants me to open up a bank account with First Tech Credit Union and give her my log in and password info. Ofcourse its sketchy!   So I questioned her and asked why she wanted that instead of her sending me the money directly from her bank to mine. She then responded because she wants to keep records of my debts. (She asked for proof and I sent her screenshots ONLY of my outstanding balances) I heard about the depositing checks scam so I questioned her about how she would deposit the funds and she said cash.  Her facebook page is 100% legit, I screenshotted her pics and family profiles etc. We came to an agreement that I would only accept her help if she sends me a picture of her ID before me sending this info.   She literally just wants to deposit the money to me, She isn’t asking me to send nobody money from that account or anything. Of course the account will be empty when I provide it to her. When she does deposit the money I am planning to immediately change the password and obviously stay watching my account activity while she deposits the funds.   She actually also provided pics of other accounts she funded. My question is, is this a scam? And if so how could she possibly scam me? Specially if i have her information and A pic of her ID. She is a christian lady and looks like she has a good heart, We talked about my struggles and she actually looked into why I needed to be assisted with the money. By no means was she pushy or anything. I want to believe that she just wants to help me but I am also too skeptical. How can I run a specific command for each find result? How would one run a specific command for each file that was found by using the find command? For the purpose of the question lets say that I would simply like to delete each file found by find. Let random vector $x = (x_1,...,x_n)$ follow multivariate normal distribution with mean $m$ and covariance matrix $S$. If $S$ is symmetric and positive definite (which is the usual case) then one can generate random samples from $x$ by first sampling indepently $r_1,...,r_n$ from standard normal and then using formula $m + Lr$, where $L$ is the Cholesky lower factor so that $S=LL^T$ and $r = (r_1,...,r_n)^T$.  What about if one wants samples from singular Gaussian i.e. $S$ is still symmetric but not more positive definite (only positive semi-definite). We can assume also that the variances (diagonal elements of $S$) are strictly positive. Then some elements of $x$ must have linear relationship and the distribution actually lies in lower dimensional space with dimension $&lt;n$, right?  It is obvious that if e.g. $n=2, m = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, S = \begin{bmatrix} 1 &amp; 1 \\ 1 &amp; 1\end{bmatrix}$ then one can generate $x_1 \sim N(0,1)$ and set $x_2=x_1$ since they are fully correlated. However, is there any good methods for generating samples for general case $n&gt;2$? I guess one needs first to be able identify the lower dimensional subspace, then move to that space where one will have valid covariance matrix, then sample from it and finally deduce the values for the linearly dependent variables from this lower-dimensional sample. But what is the best way to that in practice? Can someone point me to books or articles that deal with the topic; I could not find one. I want to uninstall VMware Player on my Ubuntu 12.04 LTS system. How should I uninstall it? How well-known is the theology of Middle-earth, in Middle-earth? One thing that is notably absent in Middle-earth is any mention of temples, churches, priests or religious rites.  There are passing mentions of Men who would "worship" Sauron, but generally the world of Middle-earth seems much less religious than medieval Europe, or even classical antiquity.  One never hears about "the cult of Aule" or "the cult of Yavanna," although historically there were many "cults of Athena" or "cults of Poseidon."  Aragorn doesn't (to my memory) make an offering to Tulkas before battle, nor does he pray to Eru Ilúvatar for victory, and all the holidays and solemnities seem to commemorate historical figures, not religious ones.  This is especially strange in a world where there are still living witnesses, like Galadriel, to the actions of the Valar and other "divine" beings, and where there are still Maiar wandering around talking to people and putting on flashy firework shows.  Now, a lot of people talk about how Tolkien didn't want the Valar to be considered "gods," but at the same time he did use phrases like "the gods of old" numerous times.  Clearly, these beings would have qualified for worship in most cultures.  And even without the Valar being considered a true pantheon, there is still Eru, a closer analogue to the Christian God of Tolkien's beliefs, and we know the denizens of Middle-earth knew of him (Him?) because of the line "Eru, who in Arda is called Ilúvatar."  And yet, Ilúvatar seems not to be terribly important to anyone on a daily basis.  So my question is this: how familiar with the "theology" of Middle-earth were the citizens of Middle-earth?  Would the average merchant in Rohan know about the Valar and the Maiar and Iluvatar and the Music of the Ainur?  The average noble in Minas Tirith?  The younger Elves, like Legolas?  The Dwarf Lords?  Were the theological truths about the formation of the world, the Valar and the Maiar just so well known that they weren't worth commenting on, or had they been forgotten by most?  And if they had been forgotten, why wasn't the knowledge (whether through deliberate teachings or casual conversation) re-invigorated by witnesses like Gandalf or Galadriel? Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing? Sum over multiple indices C++11 does not deduce type when std::function or lambda functions are involved "I want you succeeding" vs. "I want your succeeding"