I have the following problem: Let $F:K$ be a field extension, and let $E_1, E_2$ two intermediate finite fields. Let $E=K(E_1\cup E_2)$, prove that: $$[E:K]\leq [E_1: K][E_2:K]$$ Sor far, I know that $[E:K]= [E:E_1][E_1:K]= [E:E_2][E_2: K]$. Any hint is appreciated!
Suppose $K/F$ is an extension field, and $E_1$ and $E_2$ are two intermediate subfields of finite degree. Let $E$ be the subfields of $K$ generated by $E_1$ and $E_2$. I'm trying to prove that $$ [E:F]\leq[E_1:F][E_2:F].$$  Since $E_1$ and $E_2$ are finite extensions, I suppose they have bases $\{a_1,\dots,a_n\}$ and $\{b_1,\dots,b_m\}$, respectively. If $E_1=F$ or $E_2=F$, then the inequality is actually equality, so I suppose both are proper extension fields. I think $E=F(a_1,\dots,a_n,b_1,\dots,b_m)$. Since $\{a_1,\dots,a_n,b_1,\dots,b_m\}$ is a spanning set for $E$ over $F$,  $$[E:F]\leq n+m\leq nm=[E_1:F][E_2:F]$$ since $m,n&gt;1$.  Is this sufficient? I'm weirded out since the problem did not ask to show $[E:F]\leq [E_1:F]+[E_2:F]$ which I feel will generally be a better upper bound.
HowTo redirect HTTP to HTTPS on the same httpd? Here is what I have got:   CentOS 5.4 (32-bit)    installed Apache httpd (Server version: Apache/2.2.11 (Unix)) mod_rewrite already presents   Question: how to redirect simple  to  not using VirtualHost defines?  PS: tried to find in later answers on SF, but doesn't find nice solution.  Thanks. Make "compact view" as a default directory view instead of "icon view" How can I make "compact view" the default for all directory views in File Manager, instead of "icon view" ? Projectile/orbital motion over very long distance We know the optimum angle for greatest horizontal displacement when launching an object with projectile motion is 45 degrees.  How to solve the angle when it is real long distance around the earth where we cannot assume that gravity is constant but is given by $F=\frac {-GMm}{r^2}$. I use differetial equation to solve it and obtain 45 degrees.   First, for the vertical distance $y$, change $v=u-gt$ to $y'=u-y''t$. Then solve it to obtain a function $y(t)$. When $y'(t)=0$ we have the time for maximum vertical distance. Then insert the value to horizontal equation.   Can anyone guide me through this problem because I am not confident with my working Which set is unwell-orderable? In textbook it says that every well-orderable set is equipotent to an initial ordinal number. However, of course unwell-orderable cannot equipotent to any ordinal number, but is there any set is unwell-orderable? By axiom of regularity we know that every set is well-founded, so can we conclude that every set is well-orderable? If not, which set is unwell-orderable? Automate taking a screenshot of specified area Bounding lines around tax nodes I am trying to draw lines around a number of nodes. I very much like the definition of \convexpath definition given , but it is not working well on my setup (MacTex 2014 with all available updates).   I tried to compile , but what I get is the following, which is clearly not what I should be getting. Any ideas would be much appreciated. how to set camera to the same view as I turn my cursor? I want to turn the camera to get the view I want, but everytime I turn the camera, I need to go to render window to check if it's the view I want. I turn the view with my mouse cursor, but that does not affect the actual camera view in rendering, can I set the camera view the same as I turn my cursor? What does "65" mean? K-2SO beats up the troopers easily. Up in the sky, the bombers are preparing to attack. Just as they start the attack run, TIE fighters come in from the shield and swarm the rebel fleet.  JON VANDER (GOLD LEADER): Red Leader, this is Gold Leader, we’re starting our attack run on the  shield gate. Keep it tight and watch out for those towers. GOLD PILOT: More fighters. Form up, at 65.  What does &quot;65&quot; mean? Direction of friction for object that rolls with slipping Let's say you hit a cue ball with a pool stick that causes an impulse. If the pool stick hits the ball above the ball's center of mass, is the direction of friction different than if the pool stick hits the ball below the center of mass? What determines the direction? Assume the ball is rolling with slipping. Intertext vertical spacing How do I reduce the vertical spacing of \intertext? How do I remove "Ubuntu" in the bios boot menu? (UEFI) See this   I have this on my Lenovo s205 too.   Is there some way to remove "ubuntu" from the menu? I don't even have it installed anymore. Renaming user name I made a mistake in the username at the time of installation. So I changed the name of the user after logging in by creating a guest account with SUDO rights. But when I checked the list of users using ls /home command I am not seeing the new name for the user instead Old name is appearing. But I use the new name to login.   Is it like I need to change the name inside Home directory separately? Also , if there is any other changes needs to be done .. ? Why can my keyboard only digest 6 keypresses at a time? When I slam my hand onto my keyboard, only a maximum of 6 keys will be registered, I've tested this for the past few minutes and was unable to get more than 6. Is this the same on all keyboards? If not, what would be the distinguishing factor, If yes: why 6 specifically?  For clarification: I do not have a usecase for this, I am simply curious Counting unique records across 2 columns and computing a score based on boolean values I have a table that looks like this in Google Big Query:  custID  InteractionDate Purchased?  Sales 1       20150312           F         0 1       20150312           T         200 1       20150314           T         150 2       20150221           T         400 2       20150221           F         0 2       20150221           T         120   ..that I need to transform to the following structure:   custID InteractionDate Success Sales  1     20150312          0.5    200  1     20150314           1     150  2     20150221         0.66    520   ..where in this second table the success column is defined as  count(purchase=Yes)/(count(purchase=Yes) + count(purchase=No)),   and sales is just the sum.  I'm quite new to BQ and no-SQL languages. I simply don't know how to do this, I've searched and seen examples that suggest using partition over but I simply haven't seen any example that does just this. Thanks a lot in advance. I created a Compute Engine VM (Ubuntu 18.04), and when I log into it using the SSH button in the VM instances list I'm logged in as user username.  After a while, I realised I needed to be able to use another SSH client as well. As per  I enabled OS Login by adding a key-value pair in metadata, where key is enable-oslogin and value is TRUE.  I then tried to add my existing SSH key:  gcloud compute os-login ssh-keys add --key-file path/to/key.pub --ttl 0  which was successful except one major problem: it has added the key for user username_gmail_com and will only allow me to ssh to my VM as that user:  ssh username_gmail_com@myinstance  And it has also created that user on my VM. From the web interface, I'm logged in to the VM with that new user as well (/home/username_gmail_com). Of course, I can still access the original user's data (/home/username). When I disable OS Login the web interface logs me in as username again and SSH from another client is disabled.   Why is the username different when enabling OS Login? Can I change the username with OS Login enabled?   Edit  There has been an addition to the docs explaining exactly this question. Check here:  and on  all the way at the bottom under "Expected login behaviors". In electromagnetism, why does nature prefer the right-hand rule over the left-hand rule? Finding determinant for a matrix with one value on the diagonal and another everywhere else Let us look the the matrix  $\left(\begin{array}{ccccc} a &amp; b &amp; b &amp; b &amp; b\\ b &amp; a &amp; b &amp; b &amp; b\\ b &amp; b &amp; a &amp; b &amp; b\\ b &amp; b &amp; b &amp; a &amp; b\\ b &amp; b &amp; b &amp; b &amp; a \end{array}\right)$  It has one value, $a$, on the main diagonal, and another value, $b$ everywhere else. Let us assume that we are over a ring and that $a$ is invertible.  I wish to find the determinant of every $n\times n$ matrix of this form ($a$ on the diagonal, $b$ everywhere else). Using row and column operations I have managed to transform the matrix to upper-triangular form and found formula for specific cases. Generalizing it I got to the following formula:  $\det(A) = a\left(a-b\right)^{n-2}\left(a+\left(n-2\right)b-\frac{\left(n-1\right)b^{2}}{a}\right)$  I think I can prove it with row-operations in the general case with a little patience.  However, I'm wondering if there is a "smart" way of getting to this formula that I'm missing and if there is a nicer representation of it. Also, what can be said when $a$ is not invertible? (esp. the case where we are over a field and $a=0$). Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$ This is question 2.9 #9 from Topics in Algebra by Herstein:     If $o(G)$ is $pq$ where $p$ and $q$ are distinct prime numbers and if $G$ has a normal subgroup of order $p$ and a normal subgroup of order $q$, prove that $G$ is cyclic.   Let $N$ denote the normal subgroup of order $p$ and $M$ denote the normal subgroup of order $q$. Here are a couple things I noted while exploring the problem:   $N$ and $M$ are cyclic, since they have prime order. $G = NM$ [1].   Any hints where to look next?    [1] Since $N,M$ are normal, $NM \le G$. Then $o(NM) \mid o(G)$, so $o(NM)$ is either $1$, $p$, $q$, or $pq$. The first three don't work because they force one of $N$ or $M$ to be $(e)$, so we have $o(NM) = pq$. Therefore, $NM = G$. I'm new to LaTeX and trying to understand the reason why I'm getting the warning:  `h' float specifier changed to `ht'   I'm not attempting to change the float property of anything - the warning is showing up if I attempt to simply encapsulate some words in quotation marks:  ``highly distributed''   As far as I'm aware, there's no open sections anywhere and the file seems to render properly. So realistically it doesn't matter if the issue gets solved or not; but I'd like to know the reason I'm getting the warning.  Thanks in advance. I need to remove the annoying new "locking" screen that shows a giant clock: I can't remove it with just moving the mouse (as with any other desktop environment), the animation is slow, and it isn't asking me for a password!  I managed to disable it, just to discover that now I can't lock my screen on purpose, so I can't go away from the computer.  Just to clarify, I want to:   After some time the screen could go black, but if I move the mouse or press any key then I can use it directly (like a normal screensaver). If I press super-L then it is locked, and I must type the password to unlock (like a normal lock screen).   I found several "solutions", but none seems to work: