$K \leq E$ an algebraic extension. I am asked to show that each subring of $E$ that contains $K$ is a field.  I have done the following:  $K \leq E$ algebraic $\Rightarrow \forall a \in E, \exists f(x)\in K[x]:f(a)=0$  A subring of $E$ that contains $K$ is $K[a], \forall a \in E$.  Since $a$ is algebraic over $K$,it stands that $K(a)=K[a]$.  Therefore, a subring of $E$ that contains $K$ is $K(a),\forall a\in E$, where $K(a)$ is a field since it is a ring and $1\in K(a)$ so $1=f(a)g^{-1}(a)\Rightarrow f^{-1}(a)=g^{-1}(a)\in K(a)$.  Is this correct??  Or is there something I could improve??
Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?
How do that charge gets transferred while rubbing? True\False?  For every matrix $A\in M_{2}( \mathbb{C}) $ there's  $X\in M_{2}( \mathbb{C})$ such that $X^2=A$.  I know that every complexed matrix has a Jordan form matrix $J$ such that $P^{-1}CP=J$, But it's not diagonalizable for sure.  Thanks Reinstall Windows 8 on a Windows 8 based system with embedded key I'm a computer technician and we started selling few Windows 8 machine. We saw the new COA label does not show the key and after reading a little bit online, realized those systems came with the key emmebeded in the eUFI BIOS.   My question is : How do I reinstall Windows 8 if the hard drive is crashed? I kind of get that you can install it with a media and the media should activate the licence embedded. What is the media required? Is a retail DVD is able to do the work or do I need a system builder kit? I understand the manufacturer is reponsible to provide media  or replace the failed drive under the warranty but what will happen when warranty expire? I'll have to sell a full windows 8 licence with a key?  Thanks Model won't follow its armature in pose mode I've set up a rig and built it around my model, and I've linked it to my model but when I rotate the bones in pose mode, the character doesn't move with them.   My rig is the human meta rig that comes with rigify. I've seen tons of videos of it working. Why isn't it working on mine?  My character was made by creating different body parts and joining them together into the same object, then a texture was added by unwrapping them in texture mode. I'm trying to downgrade my iPhone from the current iOS to an older one, as per .  But every time I try, I always receive this error:     The iPhone could not be restored. This device isn't eligible for the requested build.     I have tried many times with and without DFU mode. Any ideas regarding this? The Problem For a small mass a distance $R_i$ away from the center of the Earth, how long would it take for the object to fall to the surface of the Earth, assuming that the only force acting upon the object is the Earth's gravitational force?  Relevant Information The following discussion seems to have solved exactly the same problem:   However, upon working out the mathematics, I'm not exactly sure how to evaluate the constant of integration.   A Partial Solution $$ F=\frac {-GMm}{s^2} $$ $$ a=\frac {-GM}{s^2} $$ $$ \frac {dv}{dt} = \frac {-GM}{s^2} $$ Multiplying by $v$ and then integrating by $dt$ on both sides, we have $$\frac {1}{2} v^2=\frac {GM}{s} +c_1$$ where $c_1$ is a constant of integration. Substituting initial conditions of $v=0, s=R$, we have $$\frac {1}{2} v^2=GM(\frac {1}{s}-\frac{1}{R})$$  At this point of time, when I use Wolfram Alpha, I get  $$c_2+\sqrt{\frac{2}{R}}t=\frac{\sqrt{s}(s-R)+R\sqrt{R-s}\times{\tan^{-1}(\sqrt{\frac{s}{R-s}})}}{\sqrt{GM(R-s)}}$$  where $c_2$ is a constant of integration. Substituting initial conditions of s=R, t=0, we find that the term $$\tan^{-1}(\sqrt{\frac{s}{R-s}})$$ is undefined. At this point, I'm stuck. Any ideas on where I've made the mistake here?  (For those interested, this question was inspired by the Greek myth which states that a bronze hammer dropped from heaven would take 9 days to hit the Earth and would reach on the tenth). What type of lens will auto-focus on Nikon bodies without a focus motor? I know my D40 doesn't have a motor in it to auto-focus lenses.  What type of lens has the motor in it? How can I know which lenses will and will not work? I want to know how EXACTLY the WPA password gets encrypted  I have been searching through and I was able to catch:   The actual password is converted to some hash Salting is applied to the hash The operation is not reversible The hash comes with the 4-way handshake   So what kind of hash is it converted? like, SHA1? MD5? and how the salting process is done? like, is the SSID appended to the password before the hashing operation?  Or maybe I have some misunderstanding here? How do I draw text at an angle using python's PIL? Using Python I want to be able to draw text at different angles using PIL.  For example, imagine you were drawing the number around the face of a clock.  The number 3 would appear as expected whereas 12 would we drawn rotated counter-clockwise 90 degrees.  Therefore, I need to be able to draw many different strings at many different angles. What exactly is "pressure", and what's its relation to force? verbatiminput and fontsize I just noticed that I lost a bunch of points from my reputation score, and I used the "reputation" tab on my user profile page to try and track down the cause.  During my investigation, I noticed there was an unusual event of type "reversal". In the normal place of a question title, it says "voting corrected".     What does this mean, and what caused it?  Did I do something wrong? Why did I lose all of that reputation? Is the system punishing me for leaving too many good answers?  Is there any way to earn it back? How to prove Vandermonde's Identity: $\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$? How can we prove that     $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$   (Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross. subequations and array in braces How do I choose a macro lens for a Canon APS-C camera? Natural Sort Order in C# Anyone have a good resource or provide a sample of a natural order sort in C# for an FileInfo array?  I am implementing the IComparer interface in my sorts. As far as we know the gaze of the basilisk will instantly kill someone or eventually petrify him/her in case the gaze is somehow softened or reflected.   But why would it melt the film in Colin's camera? The film is just plastics with some specific chemicals.   A bit broader question would be - is the some information that the basilisk's gaze is somehow destructive to non-alive objects too and under what circumstances could it happen? How to import a jar in Eclipse Lucia is gone. Just gone Skyrim problem here. I've finally found a house mod that will support a family (as in allow them to move there). I'm excited to install it, but it says that a new save game is highly recommended unless you haven't adopted any kids yet.   I recently adopted Lucia. I have two saves in my game, a quick and a full. I know that the full save has Lucia as adopted, but I'm not sure if the quick does. If it does not, then my reasoning is that I can install the mod and load that save, rather than having to work all the way back through eighteen levels.   And here's the problem. I loaded the quick save and searched for Lucia, in an attempt to see if she was adopted or not. She is nowhere to be found. She is not sitting under the tree in the middle of Whiterun, she is not in Breezehome, and she is not in the inn.   I tried to use console commands. both the moveto and placeatme commands do not work. I used both the refid and the baseid. Neither worked. I used help Lucia, thinking maybe the wiki was wrong on her ID. it gave me a different ID, which also didn't work. Every time, the console says either that it can't find the objectbaseid, or that the id is invalid.   This has led me to the conclusion that Lucia has been entirely erased from the game's memory, and I KNOW that hasn't happened. Can someone tell me what is going on?   Additional details: I am using a Breezehome overhaul mod. When I adopted Lucia, she moved in perfectly with no trouble though, so I can't really see how that would be the problem. Can I nohup/screen an already-started process?