Show $[K : F] = [K : E][E : F]$. Let $E\subset F\subset K$ be fields. Show that if $K$ is a field extension of finite degree over $F$ and $F$ is a field extension of finite degree over $E$ then $[K : E] = [K : F][F : E]$.     Since $F$ is a vector space over $E$ with degree $m$ and $K$ is a vector space over $F$ with degree $n$, then $K$ is a vector space over $F$. So $[K : E] = [K : F][F : E]=mn$.     I think I still miss some details need to show. Can anyone tell me what they are? Thanks!
Given fields $M/E/F$, why does $[M:F] = [M:E][E:F]$? Let $M$ be a finite extension of $E$ and let $E$ be a finite extension of $F$. Then $M$ is a finite extension of $F$ and $[M:F] = [M:E][E:F]$.   Is there an easy explanation and/or proof for this theorem? My instructor gave an incomplete proof in class but got stuck and I'm kind of confused of what's going on.
Vandermonde identity corollary $\sum_{v=0}^{n}\frac{(2n)!}{(v!)^2(n-v)!^2}={2n \choose n}^2$ I've been learning C#, and I'm trying to understand lambdas.  In this sample below, it prints out 10 ten times.  class Program {     delegate void Action();     static void Main(string[] args)     {         List&lt;Action&gt; actions = new List&lt;Action&gt;();          for (int i = 0; i &lt; 10; ++i )             actions.Add(()=&gt;Console.WriteLine(i));          foreach (Action a in actions)             a();     } }   Obviously, the generated class behind the lambda is storing a reference or pointer to the int i variable, and is assigning a new value to the same reference every time the loop iterates.  Is there a way to force the lamda to grab a copy instead, like the C++0x syntax  [&amp;](){ ... } // Capture by reference   vs.  [=](){ ... } // Capture copies How to install Ubuntu on a laptop with Windows 10 I want to install Ubuntu and remove Windows 10 , but when you restart the PC does not read the Ubuntu installer that is on a bootable USB. Find: $$\lim\limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right) $$  Without using L'Hospital or Taylor approximations  Thanks in advance Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. Prove  that the function $\displaystyle f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. You should give a proof that is directly based on the  definition of continuity.   I saw many similar questions on this website, but none of them gave me a satisfactory answer. I more or less understand how epsilon-delta proofs work but my problem lies more in the algebraic manipulations probably. So here's my "attempt".  Definition of continuity; for $f:A \to \mathbb{R}$     $$\forall \varepsilon &gt;0 \ \exists\delta&gt;0\ \forall x \forall x_0 \in A:|x-x_0|&lt;\delta \Rightarrow |f(x)-f(x_0)|&lt;\varepsilon$$   So from what I understand, the procedure for epsilon-delta proofs are as follows: 1. Plug the point of interest $x_0$ into $|f(x)-f(x_0)|&lt;\varepsilon$ 2. Write $x$ in terms of $\varepsilon$ 3. Then plug $x$ into $|x-x_0|&lt;\delta$ 4. Find $\delta$ in terms of $\varepsilon$. Done    At step 1, we have $\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Step 2 is where I'm stuck because of the 2nd degree polynomial in the denominator. First, I did this: $\left|\dfrac{x}{x^2+1}\right|&lt;\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Then, I wasn't too sure about what to do, I thought about getting rid of the "$1$", but that would screw up my inequality. I thought about factoring $x^2+1$, but it doesn't have any real roots. I tried inverting the fraction, but that led me to nowhere. And I've already exhausted the tools I have. How do I proceed?  Apparently, epsilon-delta proofs are the most basic building blocks of analysis. And I'm already failing, this is both embarrassing and depressing... Should actors in a game be responsible for drawing themselves? I am very new to game development, but not to programming.  I am (again) playing around with a Pong type game using JavaScript's canvas element.  I have created a Paddle object which has the following properties...   width height x y colour   I also have a Pong object which has properties such as...   width height backgroundColour draw().   The draw() method currently is resetting the canvas and that is where a question came up.  Should the Paddle object have a draw() method responsible for its drawing, or should the draw() of the Pong object be responsible for drawing its actors (I assume that is the correct term, please correct me if I'm incorrect).  I figured that it would be advantagous for the Paddle to draw itself, as I instantiate two objects, Player and Enemy. If it were not in the Pong's draw(), I'd need to write similar code twice.  What is the best practice here?  Thanks. Trying to solve     $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx  $ converges.   I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$  Would appreciate your help! I have a UK Refugee passport. Do I need a visa to go to France?   I want to go for two days with my husband, but I'm not quite sure if I can go without a visa. Hide/Show field on list form based on value from another field I need to hide field 'B' when the value in field 'A' is NOT equal to 'Other'.  This is a custom list form.  Is jquery the way to go? Changing Network Interfaces name Ubuntu 16.04 I tried to change network interfaces name on this new Ubuntu 16.04 LTS version but doesn't have the /etc/udev/rules.d/70-persistent-net.rules.  So, I tried to use /lib/udev/write_net_rules but it not exist.  Why do I need this modification? Because I'm using a tool to Simulate virtual platforms that use flex licensing and the authentication need to be in eth0 interface name.  Any suggestion?  The command ip link returns:  user@laptop:~$ ip link  1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN mode DEFAULT group default qlen 1 link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00 2: enp6s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc pfifo_fast state DOWN mode DEFAULT group default qlen 1000 link/ether b8:2a:xx:yy:xx:yy brd ff:ff:ff:ff:ff:ff 3: wlp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc mq state UP mode DORMANT group default qlen 1000 link/ether 5c:e0:xx:yy:xx:yy brd ff:ff:ff:ff:ff:ff  user@laptop:~$ ifconfig  enp6s0    Link encap:Ethernet  HWaddr b8:2a:xx:yy:xx:yy   lo        Link encap:Local Loopback   wlp7s0    Link encap:Ethernet  HWaddr 5c:e0:xx:yy:xx:yy     (Some information were ignored and suppressed)  I already tried some links: Why am I getting objects printed twice? After the training phase, is it better to run neural networks on a GPU or CPU? My understanding is that GPUs are more efficient for running neural nets, but someone recently suggested to me that GPUs are only needed for the training phase. Once trained, it's actually more efficient to run them on CPUs.  Is this true? Can you give us some way to access pending Reviews for a post from the post itself?  I think this would also help improve the accuracy and quality of reviews done, as users that come across the post naturally are more likely to be better qualified to evaluate the post, and are already taking the time to understand the post's content anyways.  In most cases, these users will be able to provide better reviews than those who are  [often quickly] going through the Review Queues.  In addition, I think it would help get more reviews done, as someone viewing a post through normal site usage would be more likely to take an interest in helping maintain that post. I know I definitely would be more active with this instead of having to go through the massive Review Queue. The task of helping fix one question that I am currently reading is far more do-able than cleaning up a large queue of questions that is largely full of content I don't care about.  I'm thinking of something like a reviews(1) link that is accessible from the post itself that appears based on your rep, or perhaps even on your rep within the post's tags. A person who cannot understand that there may be other perspectives Assuming I'm paying long term capital gains tax, what is the break even point where it is more worthwhile to switch from one fund to another?  For (a hypothetical) example, if I have a fund with an expected return of 3.5%, what would be the minimum expected return of another fund to make it worthwhile to pay the capital gains tax and switch to the other fund?  (I'm assuming the answer to this question is an equation involving investment horizon, capital gains tax rate, and my current taxable gains) How do I change my username? Some time ago, when I installed Ubuntu, I chose a rather stupid username for my account that I do not want to use anymore. How do I change this (including the name of my home directory, and the name in the terminal) without losing settings for applications? How do I keep permissions and my keys for various authentification (e.g. email, SSH, GPG and more)? What settings could possibly get lost if I changed my username? Let me add or change edit summary during edit window (grace period) prove that if $1 \le m \le n$ then $\left(1+\frac1n\right)^m < 1+\frac mn + \left(\frac mn\right)^2$ "Not a COM32R image" error when trying to install from a USB key Not a COM32R image boot   is a message that the console infinitely keeps spitting out when I try to install Ubuntu 10.04 from a USB. I created the USB using the StartupDiskCreator in 11.04.  What is the reason for this behavior? What do I do to set it right?   I have followed the steps perfectly, as enlisted in the . How to change maximum number of cores? It says in  that there is a maximum number of 8 cores supported by default in Ubuntu 12.04 Desktop/Server 32-Bit. What is the way to increase the maximum number of cores from the default of 8 to a higher value, like 12?  The reason I ask is because I have just built a machine with 6 hyper threading cores.  There were issues installing the 64-bit version of the software from a CD (it "panicked" and crashed several times).  The 32-bit installed with no problem, but I only see 8 of the 12 cores.  The BIOS is fine.  If I turn hyper-threading off and reboot, I see 6 cores as expected.  I am nervous to try and compile a kernal, I am still new to all of this.  Is there a config setting somewhere that I should change?