Suppose $u,v,w \in \mathbb{Q}$ with $u,v,w \neq 0$. Show that the determinant of the following matrix is nonzero. $$M = \begin{bmatrix} u &amp; 2w &amp; 2v \\ v &amp; u &amp; 2w \\ w &amp; v &amp; u \end{bmatrix}$$ Hint: Argue by contradiction, reduce to the case when $u,v,w$ are integers and use some number theory over $\mathbb{Z}$.  I know that the determinant is given by $$ \det M = u^{3} + 2v^{3} + 4w^{3} - 6 \,u\,v\,w $$ It is unclear to me how to utilize the hint. How to convert the problem over $\mathbb{Z}$?
I have $a,b,c\in\mathbb{Q}$ not all zero. ($a^2+b^2+c^2\ne 0$), I want to show that the following determinant is then non-zero. I failed to arrive at an appropriate form of the polynomial. Help please. $$\left|\begin{bmatrix} a &amp; 2c &amp; 2b\\b &amp; a &amp; 2c\\ c &amp; b &amp; a\end{bmatrix}\right| = a^3+2 b^3-6 a b c+4 c^3$$    Second question, what is the easiest way to argue that $\{1,\sqrt[3]{2},(\sqrt[3]{2})^2\}$ is linearly independent in $\mathbb{Q}$?    Motivation: Prove that $\mathbb{Q}[\sqrt[3]{2}] = \{a+b\sqrt[3]{2}+c(\sqrt[3]{2})^2\;|\;a,b,c\in\mathbb{Q}\}$ forms a field.   Proof: Since $\mathbb{Q}[\sqrt[3]{2}] \subset \mathbb{R}$, we prove $\mathbb{Q}[\sqrt[3]{2}]$ is a subfield of $(\mathbb{R},+,\cdot)$    $\forall (a+b\sqrt[3]{2}+c(\sqrt[3]{2})^2)\in \mathbb{Q}[\sqrt[3]{2}]\backslash\{0\}.$ We want to find $(a+b\sqrt[3]{2}+c(\sqrt[3]{2})^2)$ such that    $(a+b\sqrt[3]{2}+c(\sqrt[3]{2})^2)(d+e\sqrt[3]{2}+f(\sqrt[3]{2})^2) =$ $ (ad+2ec+2bf)+(ae+bd+2cf)\sqrt[3]{2}+(af+cd+be)(\sqrt[3]{2})^2 = 1$    Since $\{1,\sqrt[3]{2},(\sqrt[3]{2})^2\}$ is linearly independent (?) over $\mathbb{Q}$, we show there is unique solution to: $$\begin{bmatrix} a &amp; 2c &amp; 2b\\b &amp; a &amp; 2c\\ c &amp; b &amp; a\end{bmatrix} \cdot \left[\begin{array}{l l} d\\e\\f \end{array}\right] = \left[\begin{array}{l l} 1\\0\\0 \end{array}\right] $$ Which is equivalent in showing the determinant is non-zero $$\left|\begin{bmatrix} a &amp; 2c &amp; 2b\\b &amp; a &amp; 2c\\ c &amp; b &amp; a\end{bmatrix}\right| = a^3+2 b^3-6 a b c+4 c^3=(?)$$    By subfield test, 1)2)3)4) is enough to say that $(\mathbb{Q}[\sqrt[3]{2}],+,\cdot)$ is a subfield of $(\mathbb{R},+,\cdot)$ therefore a field.  EDIT: If you have shorter way that prove the proposition without touching my 2 questions, that is even better.
Scale Image Using PHP and Maintaining Aspect Ratio Is it possible to browse the internet from a laptop so that the web sites you visit think you are browsing from a mobile device?  Say I need to connect via a very low bandwidth connection and just using the mobile version of the site is sufficient?  Currently I'm using Vista and Firefox 3.5 Please stop using SoftwareEngineering.SE as your toilet bowl Stack Overflow users have been using the "off topic - belongs on SoftwareEngineering.SE" close reason as an alternative to all the other close reasons. It'd be helpful if people actually read the FAQ and the six  before voting to migrate questions like:   , no redeeming value to favorite lists that become argumentative , "help my exact situation" questions are too localized , fails 5 out of the 6 guidelines , fails 4 out of 6 , too localized and too vague , too broad   Absent a reversal of the Stack Overflow community from viewing SoftwareEngineering.SE as the Stack Overflow cesspool, can the "off topic - belongs on SoftwareEngineering.SE" close reason be removed in favor of manual migration by moderators who can actually spend the 10 seconds to determine if the question should be closed outright first?    To say that SoftwareEngineering.SE is such a nebulous place and nobody knows what its scope is as the reason why crappy questions keep getting migrated there is to miss the point. Good questions are good questions on all of the sites, and questions that are too vague, that are too argumentative, too localized, and too broad are universal close reasons. There is no special dispensation for SoftwareEngineering.SE to have bad questions, even if you personally think the questions on there suck.  Even the six subjective guidelines for subjective questions are universal guidelines: they affect all sites. So even if you had no idea what the consequences of using the "off topic - belongs on SoftwareEngineering.SE" close reason were, there are still all of the other close reasons for which bad questions, like the ones above, can be closed. $x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible? Virtualbox Windows 10 Guest installation is freezing up in 14.04 I've installed VirtualBox using the Software Centre and I'm running a system on Ubuntu 14.04 with 8gb RAM, 64-bit dual core processor.  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I am using the \let\clearpage\relax command to . The error does not occur if the file is \input instead of \included.  \documentclass[11pt,twoside]{book} %  \usepackage[utf8]{inputenc} % Enoding \usepackage[english]{babel} % Language \usepackage{listings} \usepackage{forloop}  \begin{document} \begingroup \let\clearpage\relax  \include{includedfile} % changing include for input would work  \endgroup \end{document}   Included file:   \newcounter{ct} \forloop{ct}{1}{\value{ct} &lt; 40}% {%     .\\ }  Listing \ref{mylabel}.  \begin{lstlisting}[caption=somecaption, label=mylabel] some code; \end{lstlisting}   Running this example will have an output where the label cannot be referenced (even after multiple runs):    Now, one solution is to use \input as mentioned earlier but I am rather interested in why this happens. Does the \clearpage command fulfil an important function for references to work? And can the error be avoided without removing \let\clearpage\relax while using \include?  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