Non-Relatively Prime Polynomials in a Field Extension are Not Relatively Prime in Original Field Either
Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$?
Cauchy-Schwarz inequality proof (but not the usual one) Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the inequality (for a proof of that see:  )    I am reading a book that claims the Cauchy-Schwarz inequality is actually: $$\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert$$ where $\Vert x\Vert :=\sqrt{\langle x,x\rangle}$  with the additional claim: equality holds $\iff\ x,y$ are linearly dependent  I cannot find a proof of this claim (only proofs for the dot product inner product). My question is: what is the simplest way to prove this (I define simplest to mean "using as few definitions outside of the inner product itself as possible" - so that's not including Gramm-Schmitt orthonormalisation, for example.)  I've just found some possible answers in the links on the right (annoyingly) but fortunately I have a second, albeit softer, question:  What inequalities that are common for say $\mathbb{R}^n$ are actually based on the inner product?   (In future I will check this site first, I was targeting lecture notes and such in my search)  Addendum: I want to use this on not-finite dimensional vector spaces. I've found one proof that relies on finite-ness. Capacitor with different charges on each plate I am confused as to whether/how capacitance changes when each plate has a different charge.  For example, consider a coaxial cable and put $20Q$ on the outer cable, and $-Q$ on the inner. Or how about concentric spheres, grounding either the inner or the outer? A conceptual doubt regarding Forced Oscillations and Resonance While studying about the Resonance and Forced Oscillations, I came across a graph in my textbook that is given below:-  Now, the author writes      As the amount of damping increases, the peak shifts to lower frequencies.   Why does this happen? And does this imply that at higher damping levels one cannot achieve a higher amplitude by setting the period of the forced oscillation to be equal to the natural frequency? This seems strange because, the highest amplitude is achieved when the natural frequency is equal to the driving frequency. But, I guess that the rules are somehow different for the damping case. During Windows installation I accidentally typed an accented name for my user name and my profile name is named after it. I already renamed my user to not have an accented character, but the profile folder is an other topic.  Basically I want this because I have some applications what have a problem with the accented character.  Is there a way to rename it? I know I have to make a copy from my profile, but how can I perform a relocation itself? I wouldn't like touch any other profiles on the machine just mine. How to print the entire array list without appending the index in Java? Commas that are sentence flow-stoppers ... Can I leave them out? How many Avatars have there been? Creating shapefile from current dataframe extents in layout view of ArcMap? Where is the ArcGIS 10 tool for creating a shapefile from current dataframe extents in layout view?  Have looked around and the closet thing i can find is Toolbox's Grid/Strip Map Index tools under Data Drive Pages.  I just want to be able to create a single polygon rectangle shp file based on the data frame (in layout view) for any given scale / page setup. If I want to solve the equation $4x = e^x$, is there a way to solve for $x$ without graphing and looking for intersection? bilinear form - proof I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$  is a bilinear form, that is, inter alia, the condition: $f(x+y,z)=f(x,z)+f(y,z)$  I have: $f(x+y,z)=f\left({\displaystyle \sum_{i=1}^{n}}x_{i}e_{i}+{\displaystyle \sum_{k=1}^{n}}y_{k}e_{k},{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=f\left({\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{k=1}^{n}}\left(x_{i}e_{i}+y_{k}e_{k}\right),{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=\ldots$  and what's next?  Help! How does finder sort folders when they contain digits and characters? I tried to find a topic here on this (possible problem) but I didn't.   I reported it at  but Tim Hoffmann pointed me that it is not a problem of editor. After tested with nano and pdflatex I'm posting here.  The minimal code produces errors. OK, we can solve it using \\{}.   But is this a problem with TeXlive 2014 or there are some good reasons to have that behavior.  \documentclass{report} \begin{document} test\\ [foo]  test\linebreak [foo]  test\newline [foo] \end{document}      ! Missing number, treated as zero.                        f l.3 test\ [foo] Is there a way to capture the viewport in Blender the same way as it is on 3DS Max? I know that by default I can render the viewport and everithing that is in the viewport to be the same as in the render. But, when comparing 3DS Max to Blender, I have certain limitations. In Blender I can render the view-port with vertices/edges/faces to be shown, to be selected, but not the reflections, shadows and such.  Is there a way to render the viewport in Blender as it is in 3DS Max?  Here's how I can do it in 3DS Max:  Viewport look    Render result of the viewport    Here is how I can do it in Blender:  Viewport look:      Render result of the viewport I'm putting some type on a path, a circle; but when I do it, the type is upside down. How can I flip it so it reads the right way up? See image below: First, this is about beta sites that might not even exist later, but imagine they will for the sake of this question. Taking the latest 10 questions on , I see 3 questions related to Mac OS X:    From my point of view, all 3 could also fit on , the 1st one could also be on SO, the 2nd and 3rd could also be on SU. How we will deal with this type of situation? When should we close a question as &quot;Belongs on [...]&quot;? I already see identical questions asked on 2 or 3 different sites (by the same or different users). Is there anything planned to link these questions or something like that? I mean something else than the already existing &quot;Linked&quot; feature. Something to include answers from site X in site Y or something like this. Necessary and sufficient condition for the matrix $A = I - 2 x x^t$ to be orthogonal Let $x$ be a non zeo (column) vector in $\mathbb{R}^n$. What is the necessary and sufficient condition for the matrix $A = I-2xx^t$ to be orthogonal? How to create individual chapter PDFs from included TeXs? My document structure has   one file with preamble (i.e. MainTex) and around 75 individual .tex files (i.e. Chapter_01, Chapter_02, ..., Chapter_75) representing each chapter.   I am using \include command to create a one big PDF file, the output is MainTex.pdf. Life is all good up to this point.   If I have to create individual PDF for each chapter then do I need to run 75 times with each chapter name and rename the PDF manually to match to chapter name. Because if I include only Chapter_01 and run it creates PDF named as MainTex.pdf.  Is there a way I can automate this workflow? Complete list of Lollipop system icons shown in notification bar? Is there a complete list of these? I've seen lots of Q&amp;A about "What does x icon mean" but it would be really useful if there were a complete list for reference! How can I add custom '.desktop' application launchers to Ubuntu Dock? I have made .desktop files for Ubuntu 16.04 to access my Google Chrome profiles, but once I upgraded to Ubuntu 17.10 (it worked up to Ubuntu 17.04) they got removed from the dock and I can't pin them back. Sample File: #!/usr/bin/env xdg-open  [Desktop Entry] Version=1.0 Type=Application Terminal=false Icon[en_US]=google-chrome Name[en_US]=Google Chrome Exec=google-chrome --profile-directory=Default Comment[en_US]=Chrome Name=Google Chrome Comment=Chrome Browser Icon=google-chrome Categories=Browser;Internet; Can the Dispel Magic spell be Readied to counter an opponent's spell? When playing as a spellcaster, is it valid the use of the Ready action to prepare Dispel Magic to counter a spell cast by an opponent? In the same vein, when declaring said action, do I have to specify which spell slot I will use when Readying the Dispel Magic spell? If that is the case, does this tactic nullify an opponent's spell? I get the impression that this is basically turning Dispel Magic into a spell with a casting time of &quot;1 reaction&quot;. Is my reading of the rules accurate?