Is there a prime number $p$, such that $\mathbb{Q}_p$ is isomorphic to a subfield of $\mathbb{C}$?
A few days ago I was recalling some facts about the $p$-adic numbers, for example the fact that the $p$-adic metric is an ultrametric implies very strongly that there is no order on $\mathbb{Q}_p$, as any number in the interior of an open ball is in fact its center.  I know that if you take the completion of the algebraic closure of the $p$-adic completion you get something which is isomorphic to $\mathbb{C}$ (this result was very surprising until I studied model theory, then it became obvious).  Furthermore, if the algebraic closure is of an extension of dimension $2$ then the field is orderable, or even real closed. Either way, it implies that the $p$-adic numbers don't have this property.  So I was thinking, is there a $p$-adic number whose square equals $2$? $3$? $2011$? For which prime numbers $p$? How far down the rabbit hole of algebraic numbers can you go inside the $p$-adic numbers? Are there general results connecting the choice (or rather properties) of $p$ to the "amount" of algebraic closure it gives?
What should children call a step-grandparent? My mom remarried to a great guy who has been absolutely wonderful to my kids. Ever since my mom and he were dating my kids have called him Mr. Joe. Lately my mom has been bugging me to get the kids to call him Grampa.  I agree that he deserves more respect than Mr.  The problem is that he isn't their real grampa so I don't really want them calling him that. Does anybody have any suggestions on what we should have the kids call him? They are 9 and 12 years old.  EDIT: I appreciate the comments regarding different variations of grampa and hearing how others have no issues with calling their step-granddad grampa. But I was hoping maybe someone has some suggestions for names that show the importance of the step-granddad without calling him some variation of the grampa name but still has a feeling of warmth and family in the name (I hope that makes sense). Evaluation order of Button's action Python Script to Add Fields to Feature Classes I have to try to add a field called 'Name' to a bunch of different feature classes using python, and then access the name of the feature class to fill in the field. I have no idea how to even start this! I know I need a loop but that's all I know. Apostrophes for possession: nouns My best friend of the moment - 'Essential Grammar in Use' for elementary students by Raymond Murphy - says that 'we normally use -'s for people.'     I stayed at my sister's house. (not the house of my sister).   Then, bellow in the same unit (64) says that 'we use of... for things, places, etc.'     Look at the roof of that building. (not that building's roof)   After that I thought:  Finally!!! I understand the rules for apostrophes!!!  But soon I lost all that naif excitement when I saw other written examples on the internet that were not following those 'rules'.     I've learnt that we can apply apostrophes to ...     one year's pay   and, finally, amazingly enough ... to , which means ANY kind of nouns (I deduce uncountable nouns are also part of the possibilities). This demolishes what is said in the grammar. So, it seems that eventually:     That building's roof -> is valid   So, now I need your help.    1) Can you tell me the rules concerning the apostrophe to show possession in respect to its usage with nouns? Can we really apply to all kind of nouns? e.g. teeth's, vehicle's, Paula's, forest's, manager's, Ireland's, etc     Scotland's worst drivers   Metadata improves information's accessibility       2) Is there a sense that justifies what is written in the grammar: 'we normally use -'s for people'? or it does not make sense at all? Someone's car hit our metal bar gate, so one of the rods is bent as shown in the photo. What kind of a tool can I use to bend this back, so it's straight? Thank you! How to duplicate a div in JavaScript Data from multiple lists using single REST API call What is the difference of 3 object declarations of javascript Could new users be prompted to edit instead of commenting? It's very common to ask for additional information on a question.  This is perhaps most common with new users who are a little unfamiliar with the need to put all relevant information in their question to start with.  To be blunt, I'm a little tired of explaining to new users that an edit is better than a comment, particularly where we ask "please add the output of..." I note that some users have taken to explicitly asking OPs to edit their question and adding an edit link in their comment.  But still this is missed.  More recently, adding a comment to a new user's question displays a little note saying:     ... is a new contributor. Be nice, and check out our Code of Conduct.   Could there be a similarly formatted note to new contributors commenting on their own question or answer:     If you need to add more information please  your [question][answer] instead of adding a comment. Including all the jars in a directory within the Java classpath Cross referencing between chapters of a thesis Hi I am new to the forum and fairly new to Tex, if I miss anything out please let me know and I will be glad to post it. I am writing up my PhD thesis using Texmaker on a windows system. I have my document organised with a main file and a sub-directory containing chapter 1 to 5. I want to be able to use the labels I created in chapter two for example in the appendix and vice versa. Is there an easy way to do this?   I have tried following the advice of various posts of including the commands \usepackage{xr} and \externaldocument{Chapters/Chapter_2} in the individual chapters but this leads to an error when compiling.   My Preamble is as follows:  \documentclass[12pt,hidelinks,twoside]{report} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[greek,spanish,english]{babel} \usepackage{textgreek} \usepackage[version=3]{mhchem} \usepackage[a4paper,width=150mm,top=25mm,bottom=25mm,bindingoffset=6mm]{geometry} \usepackage{amsmath, amsfonts, pifont, float, color, url} \usepackage[pdftex]{graphicx} \usepackage[style=numeric,backend=biber,sorting=none]{biblatex} \addbibresource{references.bib} \usepackage[pdftex, plainpages=false, pdfpagelabels, bookmarks=true]{hyperref} \usepackage{fancyhdr} \usepackage[center,font={it,small}]{caption}   I then have a section such as this in the main document:  \begin{document} %%... Title page ... \chapter{Theory and Methods} \label{chap:Theory and Methods} \include{Chapters/chapter_2} %%... Other chapters .... \end{document}   In Chapter 2 I have for example a label for a diagram which I may want to reference in other chapters:  \begin{figure}[H] \centering \includegraphics[height=10cm,width=\textwidth]{RF_colour_hi_res.pdf} \caption{Random Forest} \label{fig:2.1 Random Forest} \end{figure}   How can I have this label show up in other chapters after I type \ref ?  Many thanks in advance.  If any other information is required please let me know.   This has been answered in the comments although the references do not appear in the auto complete if they are copied and pasted in they still work. What does “{} \;” mean in find (in Linux)? Sometimes I see a command like  find . -name * -exec ls -a {} \;   I was asked to execute this.  What does {} \; mean here? How many numbers are in the Fibonacci sequence below a given threshold? Assuming I'm asked to generate Fibonacci numbers up to N, how many numbers will I generate? I'm looking for the count of Fibonacci numbers up to N, not the Nth number.  So, as an example, if I generate Fibonacci numbers up to 25, I will generate:   1, 1, 2, 3, 5, 8, 13, 21 that's 8 numbers   How do I calculate this mathematically for an arbitrary "n"? How to compute the marginal likelihood given two multivariate Normal distribution? Given that $a$~$\mathcal{N}(\mu, \Sigma)$ and $\mu$~$\mathcal{N}(0, \sigma^2 I)$, Is $p(a|\Sigma, \sigma)=\int p(a|\mu, \Sigma)p(\mu|\sigma)d\mu$ again a Normal distribution? And how to estimate the mean and variance respectively? Thanks a lot! Provide two arguments to one option using getopts In below code when I give option r then getopts requires one arguments:  while getopts ":hr::l:" opt; do     case $opt in         r ) echo "Run Numbers - argument = $OPTARG " ;;         l ) echo "Latency range - argument = $OPTARG" ;;         h ) helptext             graceful_exit ;;         * ) usage             clean_up             exit 1     esac done   But I need to pass two arguments after -r option, instead of one. Is there an easy way to do this? A toplogical space $X$ is said to be second countable   if there exists a countable basis for the topology.      $X$ is separable if there exist a countable dense subset.   Show that a second countable space always is separable.      Give an example of a space that is separable but not    second countable   The first part was easy and i managed to solve it on my own  Assume that $X$ is second countable and let $B_1, B_2, B_3...$ be a countable basis. Construct the set $A$ by taking one element $x_n$ out of every $B_n$. Then $A$ becomes dense because every open set $U$ contain a $B_n$ which in turn contain the element $x_n$ in $A$.    Again, like in the question i posted yesterday, i had trouble to come up with an example but our professor gave an example during the last lecture but i dont fully understand it:  "For an example of a separable but not second countable space let $X$ be the real line provided with the cofinite topology, i.e. where a set is open if its either empty or if its complement is finite.   Then every infinite set is dense and hence $X$ is separable.  But for an arbitrary countable family of open sets $U_1, U_2...$ write $U_i = X$\ $A_i$, where $A_i$ is finite, there is an element $x \in X$\ $(\cup_i Ai)$, because $A_i$ is countable,  such that the open set $X$\ $\{x\}$ is then not able to contain any $U_i$. Consequently $X$ is not second countable."  We asked him to explain it in more detail but his english is poor and nobody understood anything...:)       Def: A subset $A$ of a topological space $X$ is dense in $X$ if for any point $x$ in $X$, any neighborhood of $x$ contains at least one point from $A$.   $1.$ "Then every infinite set is dense"  Take $A$ to be one of these infinite sets. If $x \in A \subset X$ then it trivially fulfills the definition. But if we take a $x \not\in A$ why is it not possible to find a neighborhood in these finite sets that does not contain an element in $A$?  edit: its obvious if it is only open neighborhoods, open sets in the topology, but that is not specified in the definition.  $2.$ I am not fully understanding this either  "But for an arbitrary countable family of open sets $U_1, U_2...$ write $U_i = X$\ $A_i$, where $A_i$ is finite, there is an element $x \in X$\ $(\cup_i Ai)$, because $A_i$ is countable,  such that the open set $X$\ $\{x\}$ is then not able to contain any $U_i$. Consequently $X$ is not second countable."  Is it possible to express this in a simpler way? I recently came across an IE7 only bug that I thought I'd share so when I come to this site 6 months from now to figure out the same thing, I'll have it on hand.   I believe the easiest way to recreate this bug would be the following html in a page with a declared doctype (it works correctly in "quirks mode" / no-doctype):  &lt;div style="overflow: auto; height: 150px;"&gt;     &lt;div style="position: relative;"&gt;[...]&lt;/div&gt; &lt;/div&gt;   In IE7, the outer div is a fixed size and the inner div is relatively positioned and contains more content (assuming the inner div causes an overflow). In all other browsers, this seems to work as expected.  Screenshot: Prove that $a^{2^n}=1 \mod 2^{n+2}$ I would like to prove that  $$a^{2^n}\equiv 1 \pmod {2^{n+2}}$$  I tried induction but could not get it.  Thank you very much! I want to find the interval of convergence of $$\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(2x)^n$$ By the root test, $\sqrt[n]{|(\frac{n}{n+1})^{n^2}(2x)^n|}=|(\frac{n}{n+1})^{n}(2x)|$  And $(\frac{n}{n+1})^n=(1-\frac{1}{n+1})^n\to\frac1e$ as $n\to\infty$  So I've got an open interval(of convergence) s.t. $|\frac1e 2x|&lt;1\implies |x|&lt;\frac e2$.  Now I need to figure out whether the series is convergent or not at the end points, $x=\frac e2, -\frac e2$.  So there are two series I need to deal with, namely, $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}e^n$ and $\sum_{n=0}^\infty (\frac{n}{n+1})^{n^2}(-1)^ne^n$.  Suppose $(\frac{n}{n+1})^{n^2}$ is increasing. Then $$(\frac{n}{n+1})^{n^2}&lt;(\frac{n+1}{n+2})^{(n+1)^2}\iff 1&lt;(\frac{n+1}{n+2})^{(n+1)^2}(\frac{n+1}{n})^{n^2}$$ $$\iff 1&lt;(1-\frac{1}{n+2})^{2n+1}(1+\frac{1}{n^2+2n})^{n^2}$$  But $(1-\frac{1}{n+2})^{2n+1}\to\frac{1}{e^2}$ and $(1+\frac{1}{n^2+2n})^{n^2}&lt;(1+\frac{1}{n^2})^{n^2}\to e$  So I have $1&lt;\frac1e&lt;1$, which is a contradiction. Hence $(\frac{n}{n+1})^{n^2}$ is not increasing. But I don't know if it's decreasing. (is it true that $(1+\frac{1}{n^2+2n})^{n^2}\to e$?)  If $(1+\frac{1}{n^2+2n})^{n^2}\to e$ is true, then $(\frac{n}{n+1})^{n^2}e^n$ converges to some positive number, so $\sum(\frac{n}{n+1})^{n^2}e^n$ diverges. But I don't know if it's decreasing: can't apply the alternating series convergence test(Leibniz's criterion).  So, can you help me with the two end points? I started to work with EEVEE short time ago and I was amazed by the capabilities of this engine. However, I noticed something strange when I was rendering some tests. Some objects/reflections/shadows dissapear when they're off canvas. I imagine that maybe this is for save memory data, but it can be not suitable to some projects. To make my point graphical, I made a simple scene with simple objects. The main object is the sphere with a mirror-like shader. Out of focus there are different objects with different shaders. A yellow cone, a blue cylinder and a white cube light. But surprisingly, those objects dissapear when the composition is rendered. See the rendered animation below:  Notice that the blue cylinder is not even showing at all in the final render. And the blooming effect from the light cube also dies. I would like to improve this bahaviour and render the objects that are still out of canvas using EEVEE. Do I missing something like memory configuration or reflection/shadow tweaks? Is this caused by the EEVEE engine or by my GPU config (GTX1060)? Thank you in advance!