Let $p$ and $q$ be prime numbers, $p\neq q$.  Now I have to show $L:=\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})=:K$.  The direction $K\subseteq L$ is clear. For the other direction I have to show that $\sqrt{p}\in K$ and $\sqrt[3]{q}\in K$. Thats where I'm at at the moment.   Can someone please give me a hint on how to show this?
Can someone please give me a tip how to show that for distinct prime $p,q$ that $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ and how to find the minimal polynomial of $\sqrt{p}\sqrt[3]{q}$ over $\mathbb{Q}$ ?
What's the best way of checking if an object property in JavaScript is undefined? How to install Windows 7 after Ubuntu and dual boot? How can I maximize the "blurry background, sharp subject" (bokeh) effect? Find a value anywhere in a database Preposition confusion - Do you learn somthing 'at' school OR 'in' school? I know that we use the preposition at to describe a place. I learned French at school is one of the examples I can think of.   I came across two examples that use the preposition 'in' with 'schools' in the context of 'learning'.   Example #1 from the  website:      Eleven things that you will never learn in school.   Example #2 from      Do you think the skills you learn in school will help you get the job you want?   The use that I had in my mind is here on      Should Parents Control What Kids Learn at School? By "hand cipher", I mean a symmetric cipher for which encryption and decryption can can both be performed with a pencil on graph paper, consuming about 10-20 seconds per character by a proficient user. Additional simple tools, e.g. a deck of cards or a multiplication table, may be used to expedite the process. No mathematical proficiency is required other than mental addition, subtraction, and maybe multiplication.  Basically, it should be a cipher that someone of average mathematical abilities can master, and it should require no digital technology or cryptographic tools that cannot be constructed out of paper.  For example, four-square in CBC mode seems secure to me because CBC is a well established mode of operation and four-square seems to be a good block cipher as it accommodates large keys and seems to produce a random-looking output. However, the fact that four-square in CBC was never widely adopted before digital encryption became a consumer technology leads me to believe that someone broke the cipher and thus discouraged people from using it.  So is there any hand cipher that the best supercomputers would take years to cryptanalyze? If not, then for which hand cipher is the best known attack the most complex? Nikkor f/1.8G 35mm or 50mm? I'm new to photography. I got a Nikon D3100 with the 18-55 kit lens.  I was looking to buy a f/1.8 prime lens, and I found two options:   AF-S DX NIKKOR 35mm f/1.8G   AF-S DX NIKKOR 50mm f/1.8G   How should I go for choosing one or the other? Possible Duplicate:          This is something I haven't quite figured out yet, but I have been using function(){}() just because my VIM syntax highlight screws up if I add the parenthesis, although I've seen (function(){})() around many times, maybe its an IE thing?  edit:  var singleton = function() {     // code }();  var singleton = (function() {     // code })(); Firstly my main goal is to try and get FTP files recursively in Centos.  However the actual FTP login contains an @ which ruins the syntax.  Is there some way to place the username in "double quotes" or some such equivalent such that I can use @in the username? Opening PDF in Illustrator breaks up text objects I recently had to convert some PDF files to SVG, and did this by opening the PDF in Illustrator, and saving out to SVG.    The issue was that when Illustrator opened the pdf, many (but not all) text objects would be broken up into different text objects.  For instance, the word "policy" wouldn't be one text object, but rather several text objects, "po", "l", "icy".  There didn't seem to be any rhyme or reason to it.    How can I prevent this from happening? Which of these two phrases has more negative undertone?      "The interview I had was not very good"    or      "The interview I had was not good" How can I report bugs with or request features for the Stack Exchange mobile apps? I'm using the official Stack Exchange mobile app for iOS or Android, and  My app crashed! It doesn't look good on my new smartphone. It won't run after I updated my device. I noticed another bug. I have an idea for a new feature which is currently not in the app.  How do I report bugs with or request features for the iOS and Android apps? Also:  I've used the built-in functionality of the app to report a bug or submit a feature request. Did I do the right thing? The app prompted me to post on Meta after an error occurred. Should I post about it here? I'm wondering why the latest version is from 2017. I've heard rumors that the apps are no longer supported; is that true? I can't find the app on my device's app store anymore! Where did it go?     We get a lot of bug reports from users about the mobile apps, which are unsupported, and at the moment, there is no centralized Q&amp;A pair for those to be closed as duplicates of (the relevant answers are spread across multiple unrelated questions). This question is an FAQ proposal, to serve as a directly-related duplicate target for those questions. Workflow for converting LaTeX into Open Office / MS Word Format I often have to write up reports based on the analysis of some data. I use R to analyse the data and export tables, figures, and text. This is then included into a LaTeX document either using input or Sweave ().  However, when I collaborate with others, I sometimes need to provide a document in Open Office / MS Word format.  Question:  Thus, assume the simplest scenario   I have a LaTeX document with text, tables, and figures I need to export this reliably into Open Office or MS Word format: this includes mathematical formulas, table formatting, and quality figures) I don't need to go back from MS Word to LaTeX   What is a reliable, efficient, and preferably free process?  Initial Thoughts:   Export to HTML,  LaTeX - LyX - ODF Use odfWeave  that deals with some specific issues associated with LaTeX to Open Office conversion    I was hoping that there is an expert out there who has worked out a good system already. Making voltmeter accepting bipolar input voltage, using a microcontroller I want to build a very simple voltmeter using ATMEGA328 ADC. I can successfully measure up to 20v (using a voltage divider), but the problem is, I don't understand how to measure the voltage properly, if negative and positive terminals are switched.   Only way I could think of is adding another ADC input which is responsible for measuring the 'inverted voltage'. The join both ADC inputs, but add diodes in between, so one channel works with (+-) and the other channel works with (-+) and then 'merge' the input from both channels using code. Then I should be able to measure things like sin wave for example (where polarities switch).   Is the concept correct or is there a better way to do this? Thanks! I use bash on both my Mac and as my cygwin shell. Overall, I'm quite pleased with it.  My one complaint however is that its TAB completion isn't as good as Windows Command Line TAB completion.   For example, if there are two files with the same prefix, then bash just beeps at me when I try to TAB complete. If I keep nailing TAB, then it will eventually show me a list of potential files.  I definately prefer how Windows will cycle through potential files right on the command line.   Is there a way to get bash to operate similar to Windows Command Line in this regard?  If there isn't (I do not want to switch shells), am I missing something about TAB completion in bash? Is there a more efficient way to use it? I'm looking for a symbol for skeptical consequence so that I can write that something, B, is a skeptical consequence of something else, A.  In a book I read it's written like \(A \vdash B\) or \(A \sststile{}{} B\) but the horizontal line is wavy like \sim, has the same width as the vertical line and does not vary in width. So, what I'm looking for is a binary relation symbol that is similar to the combination of the first two and the last relation in the following image:    That is, something similar to the following mockup but the wavy line should be such that it has a constant width and the same width as the vertical line:    I've tried  without success. Testing whether two regression coefficients are significantly different (in R ideally) If this is a duplicate question, please point to the right way, but the similar questions I've found here haven't been sufficiently similar. Suppose I estimate the model $$Y=\alpha + \beta X + u$$  and find that $\beta&gt;0$. However, it turns out that $X=X_1+X_2$, and I suspect $\partial Y/\partial X_1 \ne \partial Y / \partial X_2$, and in particular, that $\partial Y/\partial X_1 &gt; \partial Y / \partial X_2$. So I estimate the model $$Y=\alpha + \beta_1 X_1 + \beta_2 X_2 +u$$and find significant evidence for $\beta_1,\beta_2&gt;0$. How can I then test whether $\beta_1&gt; \beta_2$? I considered running another regression $$Y=\alpha +\gamma(X_1 - X_2) + u$$ And testing whether $\gamma&gt;0$. Is this the best way?  Also, I need to generalize the answer to many variables, i.e. suppose we have $$Y=\alpha + \beta^1 X^1 + \beta ^2 X^2 + \dots + \beta^nX^n + u$$ where for each $j=1,\dots,n$, $X^j=X_1^j+X_2^j$, and I would like to test for each $j$ whether $\partial Y/\partial X_1^j \ne \partial Y / \partial X_2^j$.  By the way, I am primarily working in R. I have a JavaScript widget which provides standard extension points. One of them is the beforecreate function. It should return false to prevent an item from being created.   I've added an Ajax call into this function using jQuery:  beforecreate: function (node, targetNode, type, to) {   jQuery.get('http://example.com/catalog/create/' + targetNode.id + '?name=' + encode(to.inp[0].value),    function (result) {     if (result.isOk == false)          alert(result.message);   }); }   But I want to prevent my widget from creating the item, so I should return false in the mother-function, not in the callback. 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