Find three integers $n_1, n_2, n_3$ such that $n_1^3 + n_2^3 + n_3^3 = 42$
Integer solutions to the equation $a^3+b^3+c^3=30$
How to prove a finite field is not ordered? I have a set S={0,1}, and the addition and multiplication rules are  \begin{array}{c|cc} +&amp;0&amp;1\\ \hline 0&amp;0&amp;1\\ 1&amp;1&amp;0 \end{array}  \begin{array}{c|cc} *&amp;0&amp;1\\ \hline 0&amp;0&amp;0\\ 1&amp;0&amp;1 \end{array}  It is sure that it is a field. How can I prove this field can be ordered or not? How do you prove the follwing $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$ how do you prove this inequality? $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$ At first glance I thought the triangle inequality would be useful but it's not form what I see. If I raise everything to the power of n I get the following: Possibility 1 (n uneven) : $|x + ... -y| \le |x-y|$ I guess I would have to prove that ... $\le$ 0 but I don't know how to do that. Possibility 2 (n even) : $|x + ... +y| \le |x-y|$ Same as above but now I have to prove that ... $\le$ -y. Am I even on the right track here? Any help would be appreciated. Thanks in advance. How to move boot and root partitions to another drive Are there any banned subjects on Area 51? I know of the following rules for creating a proposal on Area 51:  The scope must be reasonably well-defined, not too unclear (e.g. &quot;cool stuff&quot;), too opinion-based (e.g. &quot;car reviews&quot;), too broad (e.g. &quot;life on Earth&quot;), or too narrow (e.g. &quot;The 1995 academic misconduct scandal at East Oak Mews High School&quot;). One  come to Area 51 with an already existing community rather than hope to build one on Area 51. Proposals that are subsets of existing sites are not acceptable (e.g.  — deleted). These get closed as &quot;This subject is already well-covered by a live Stack Exchange site. We generally do not split off subjects simply to give them their own space.&quot; Proposals for non-English sites .  Other than the criteria above, are there any per se restrictions on site content for an Area 51 proposal? If so, what topics are currently banned? I would expect a proposal for a &quot;Let's organize a lawsuit against Stack Exchange&quot; or &quot;[Specific SE staff member] sucks, amirite?&quot; would get closed for obvious reasons, but I'm interested if there are any non-obvious topics that are not currently allowed in proposals regardless of how well-defined, well-scoped, original, and well-supported by a community they are. Having a list of banned topics would be helpful to potential proposers as well as the communities that back them. For example, I wouldn't want to go to all the effort of organizing a community, bringing them to Area 51, etc., only to have my proposal closed with a curt &quot;Sorry, due to the political climate, we are not currently accepting proposals for sites about real-life weapons.&quot; custom closure. To be clear, I am asking solely about topic restrictions (e.g. &quot;no sites about fishing, hunting, or weapons&quot;). I'm not asking about clarity requirements, the requirement to have a pre-existing community, not duplicating an existing site, or the requirement to not use sock puppets to artificially inflate proposal stats. Those requirements are already well-defined elsewhere or else are common sense. Also to be clear, I don't have a specific proposal idea in mind right now or the community to make it happen. I'm asking in general. The question could be rephrased as &quot;What are those topics where I shouldn't even bother to try to define, gather a community, etc. because any attempt to propose a site will be summarily closed due to covering a banned topic?&quot; This is not a duplicate of  because that question is about the appropriateness of expanding a proposal to cover things that are acceptable proposal topics but arguably don't make sense in combination with the original proposal scope. My question is about topics that are completely banned from proposals regardless of whether they are included in the original proposal description or tacked on by the community later. Typesetting 144...4 with "n times" under the 4's is easy, but what about \sqrt{144...4}? Inspired by  on the math StackExchange, suppose you want to typeset the square root of the number 144...4, that is, 1 followed by n 4's. If you write $\sqrt{1\underbrace{44\ldots4}_{n\ \text{times}}}$, as in the current version of the question, you get a huge radical sign that reaches down to match the "n times" text below the number. Is there an elegant way to typeset this so that the radical sign looks like it would with $\sqrt{144\ldots4}$, but also keeping the underset text in the right place? Mods folder missing from .minecraft folder I have Windows 10 and when I go to look in my .minecraft folder in the AppData folder, the mods folder is not there.    How can I find the mods folder? crm 2011 how to hide/show the ribbon button with javascript Concatenating two std::vectors Is $0$ a natural number? Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?  It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers. Office 2016 Advanced Setup Beamer keeping line position constant on itemize I want to have a slide in beamer with itemize exposing new line on each click while at the bottom part of the slide there are some equations. This is my attempt  \begin{frame}   \begin{overprint}{\textwidth}     \leavevmode     \begin{itemize}[&lt;+-&gt;]     \item &lt;1-&gt; Poisson Boltzman equation (PB)     \item &lt;2-&gt; Debye-H\"uckle equation  (DH)     \item &lt;3-&gt; The naive approach - applying (DH) equation to PE     \end{itemize}   \end{overprint}   \begin{tikzpicture}     \node at (0,0) {};         \node&lt;1,2&gt; [text width = 20cm]at (10.5,-1) {$ \nabla^2 \phi =\sum_i \frac{n_i q_i\text{e}^{-q_i \phi/kT}}{4 \pi \epsilon}$};         \node &lt;2&gt; [text width = 20cm]at (10.5,-2) {$q_i \phi/kT \ll 1 \Rightarrow \nabla^2 \phi \simeq \kappa^2 \phi$};         \node &lt;2&gt; [text width = 20cm, font=\footnotesize]at (10.5,-2.5) {$\kappa^2 = \lambda^{-1/2} =  \sum_i \frac{n_i q_i}{4 \pi \epsilon}$};         \node &lt;3&gt; [text width = 20cm] at(10.5,-1) {$\frac{F_{el}}{VkT} = - \left ( \frac{ 4 \pi e^2}{\epsilon kT}     \sum_{i=1} n_i z_i^2 \right )^{3/2} \left ( 12 \pi \right )^{-1}$};   \end{tikzpicture} \end{frame}   However, the itemized line jump up and down between slides. I've tried to use itemize without any environment surrounding it. I've also tried to use overlayarea instead of overprint. How can I keep the line position constant?  I've uploaded a gif animation to display this. On my machine it run really fast and I can't fix this. Hope the problem is understood.   When I run this, I get How to include a PDF image into a PDFLaTeX document Images as Planes in Eevee having weird transparency How do I prove $F(a)=F(a^2)?$ Let $E$ be an extension field of $F$. If $a \in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$.  let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and $k$ be the degree of the minimal polynomial $q(x)$ of $a^2$ over $F$.   Since $a^2 \in F(a)$, We have $F(a^2) \subset F(a)$, then $k\le n$  In order to prove the converse:  $q(a^2)=b_0+b_1a^2+b_2(a^2)^2\ldots+b_k(a^2)^k=0$   implies   $q(a)=b_0+b_1a^2+b_2a^4\ldots+b_ka^{2k}=0$ Then   $p(x)|q(x)$, because $p(x)$ is the minimal polynomial of $a$ over $F$.  If I prove that $n|2k$ we done, since $k$ is odd, we have $n|k$ and $n\le k$ and finally $n=k$.  So I almost finished the question I only need to know how to prove that $n|2k$ It should be only a detail, but I can't see, someone can help me please?  Thanks Apparently, my linux ubuntu setup of QGIS doesn't contain grass gis, although when I initially installed it last week I added grass gis to the command line. Anyway, I typed in once again and installed grass gis 7.0, however while GRASS starts, the plugin QGIS for it doesn't show up. Any ideas? Recommendations for good Newtonian mechanics and kinematics books What are some good books for learning the concepts of Kinematics, Newton laws, 2D Motion of Object etc.? After a brief discussion with @MonicaCellio (In the comment) - It has come to my attention that perhaps I chose the wrong words to describe my original idea (In the edit history). When I wrote "social features" I was actually suggesting an alternative way to for users to share their knowledge using a new type of post and articles and not just by questions and answers.   The basics of my idea was to allow users to write articles on topics relevant to the community on the SE websites just as any answer posted is expected  to be too. Allowing users to write their own content (I.E posts/articles/etc..) will be granted as a privilege for advanced users so that not anyone could write stupid things, users that will write content must be established and well-familiar with the rules of SE.   I know anyone can start a blog, but it's require frequent maintenance and content updates, while writing on SE ensure that it addressed to a large, well-established community, with a similar fields of interest.  Posts writing will follow the same guidelines as asking and answering questions does, but it will provide an alternative for users that are mostly answering specific questions, as a new method to contribute to the community.   By writing, one may also reveal more about their personality, wit, way of thinking, ETC. That's when the "social" takes place - You'll also have "authors" and not just "users" you can remember by their catchy name, but people that you can remember by the field of interest they are writing content about.   They key points will be:    Users will be able write rich content with advanced WYSIWYG HTML editor, content that doesn't require a question to answer, but that is still relevant to the SE website it is writing on. Users will be able to Upvote/Downvote other users posts. Posts will be listed on their author profile, so it will not just show the activity, but additional layer of user generated content. Popular posts with high reputation could be associated to tags making the information of tags richer with user generated content Users will be able to include links in answers not only to other questions, but also to posts. User will be able to comment on post to allow the author to correct and improve it. It will improve the interaction and offline discussions (Which are prohibited when it come to long discussions on questions and answers). I believe that users will be happy to share their own tutorials, tips and tricks - Because they are already doing so when answering questions, but before the can start, they'll have to earn enough reputation to ensure that only high profile users will be able to write high profile content, and not just any rubbish. It will provide another type of method for voting and getting earning reputation Users will be able to include content in their CVs on SO Jobs.  Opportunity to earn some new badges.   So it's a Mega feature request and it require some detailed design work, but I do think that it will invite a new type of activity for users on SO (and the rests of the sites of-course). Can I enable AHCI after installing Windows 7 in a SATA drive that was on IDE mode? Every ordered field has a subfield isomorphic to $\mathbb Q$? I'm going through the first chapter in a text on real analysis, which contains preliminaries on ordered fields, the real numbers, etc. Supposedly I had learned about such things already, in calculus, but I thought it wouldn't hurt to go over it again.  Up until the paragraph which is the subject of my question, everything is thoroughly proven and examples are provided. However the following excerpt just goes through facts, whose proof I cannot conceive:      Although $\mathbb Q$ is an archimedean field, these properties cannot be used to define $\mathbb Q$ since $\\$ there are many Archimedean ordered fields. What distinguishes $\mathbb Q$ from the other $\\$Archimedean ordered fields is that $(\mathbb Q,&lt;)$ is the $smallest$ ordered field in the following sense: $\\$ if $(X,\prec)$ is an ordered field, then $X$ contains a sub-field which is (field) isomorphic to $\mathbb Q$.$\\$ Furthermore such an isomorphism preserves the order relation.   I've concatenated the paragraph a bit, so the text isn't idem from the book (if anyone wishes to know it's Phillips, An Introduction to Analysis and Integration Theory, Dover Publications). Also $\mathbb Q$ contains no proper subfields, and this can be verified by the apparently easy-to-prove fact that, if a field has characteristic zero, then it contains a subfield isomorphic to $\mathbb Q$. Then it remains to prove the preservation of order.  I'm guessing that some concepts from abstract algebra or field theory would easily suffice, but at the moment such topics are a bit over my head, so all I can think of doing is actually constructing the isomorphism $\phi:\mathbb Q\to\hat{\mathbb Q}$, where $\hat{\mathbb Q}$ is a certain subfield of  $X$ in such a way that field operations are preserved, but I really can't come up with anything. So the question is: how do I construct this isomorphism, or, if there's a better way of proving this, how is it done? Thanks for any help.  Edit: Assume the existence of $(\mathbb Q,&lt;)$ as an archimedean ordered field. Let $S = \{ax + by: x,y \in \mathbb{Z}\}$  and let $e &gt; 0$ be the smallest element in $S$. Prove that $e \mid a$, and hence prove that $e = \gcd(a,b)$  I'm afraid I can't provide much of my initial working here because I'm at a loss of how to start.  I suppose that to show $e\mid a$ I would have to show that $eg = a$ for some $g \in \mathbb{Z}$. But I have no idea how I would do this.