Seems intuitive. Like the fact that $\sqrt 3 \not \in \mathbf{Q}(\sqrt 2)$. But how to approach actually proving it? The proof of this fact doesn't seem to generalize well.
I need to find a way of proving that the square roots of a finite set    of different primes are linearly independent over the field of    rationals.    I've tried to solve the problem using elementary algebra  and also using the theory of field extensions, without success. To  prove linear independence of two primes is easy but then my problems  arise. I would be very thankful for an answer to this question.
How does the radius of a pipe affect the rate of flow of fluid? Poiseuille's law states that the rate of flow of water is proportionate to $r^4$ where $r$ is the radius of the pipe. I don't see why.  Intuitively I would expect rate of flow of fluid to vary with $r^2$ as the volume of a cylinder varies with $r^2$ for a constant length. The volume that flows past a point is equal to rate of flow fluid, and thus it should vary with $r^2$.  I cannot understand the mathematical proof completely. Is there an intuitive explanation for this? Is there a way to block a list of specific apps on macOS?  I have looked up solutions such as Cisdem, but there are some holes in the way it works that I don't like.   Unless you also block Activity Monitor, Terminal, and System Preferences, you can simply quit the Cisdem process or create a new user to get around it. You have to block Activity Monitor, Terminal, and System Preferences to make it work well.   I want to block an app from running or even being installed on my Mac. The app can be installed via the App Store and the web. Not sure how to go about this. Enabling Parental Controls won't work as Administrator privileges are needed.   The specific app I want to block is Apple Configurator.  Any ideas? Hide c# windows application source code I wrote a windows application using C# .Net 2.0 and i want to do something which hide the source code, so when any one use refactor tool can't see the source code. I used dotfuscator but it just changed the function names but not all the source code.  UPDATE: I want to hide the source code, not because of hiding the key, but to hide how the code is working.  Thanks, Execute command in remote active terminal How do I create an Excel (.XLS and .XLSX) file in C# without installing Microsoft Office? How can I create an Excel spreadsheet with C# without requiring Excel to be installed on the machine that's running the code? I recently bought the whole PC series of Saints Row and after I began the third, after I leveled up, I saw that most of the power upgrades were already done for me and many clothing items were already bought.  I know the game is quite easy  economically but I want to start a game in which I can actually level up, not sideways.  Is it possible to begin a game in which (preferably) everything is locked?  I probably have all the downloadables of the game. I am trying to find all the Sylow 2 subgroups of S4 using Sylow’s theorems. Now, I know that a Sylow 2 subgroup of S4 has size 8, and that there are either 1 or 3 of them (as the number of of Sylow 2-subgroups has form 1+2k and divides 3, the index).  Now my lecturer states “stabilisers of the 3 different bisections of {1,2,3,4} yield 3 distinct Sylow 2 subgroups”  Now my questions are:  1) What does it mean by stabilisers of 3 different bijections of {1,2,3,4}? Is that to say the elements of S4 which send {1,2} and {3,4} to {1,2},{3,4} i.e. by (12)(34)?  2) How does he know that there are 8 elements of this set? Perhaps this will become clearer after the 1st question is answered.  Many thanks, group theory is hard. How to make DNS server propagate faster if you are a site owner? I need to rename a bunch of files in order to fix their sort order, therefore I need to be able to do a sort of "find and replace" so I can replace chunks of the filenames at a time.  What tools should I use? I prefer GUI but a command line tool recommendation would also be fine. Limits involving factorials $\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$ I am trying to calculate the following limit  $$\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$$ where $k$ can be any number between $0$ and $N$.  I thought of the following:  If I take the logarithm of the expression then I get:  $$\lim_{N\to\infty} \left(\log(N!)-\log((N-k)!)-k\log N\right)$$    Using the Stirling formula this can be approximated as:  $$\lim_{N\to\infty} \left(N\log(N)-(N-k)\log(N-k)-k\log N\right)$$  Now there are two cases: If $k$ is $N$, then the second term vanishes and the remaining terms cancel. If $k$ is smaller than $N$, then I can drop the $k$ inside the second logarithm and all the terms cancel. So the limit $$\lim_{N\to\infty} \log\left(\frac{N!}{(N-k)!N^{k}}\right)=0$$ Which means that: $$\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}=1$$ I don't know if this is mathematically rigorous. Would like some help. Thanks Find the first occurrence/starting index of the sub-array in C# Correct usage of "lest" Which of the followings is/are the correct usage of the word "lest"? How are these different from each other?   ...hesitant to speak out lest he be fired. ...hesitant to speak out lest he'd be fired. ...hesitant to speak out lest he would/will be fired.   P.S. I took point number 1 as the reference for this question from the definition of lest from the Merriam-Webster Dictionary. Black flakes from new cast iron pan I bought a brand new Lodge 12" cast iron pan, washed it in soap and warm water and dried it with a paper towel thoroughly. Then I applied a thin layer of canola oil with a paper towel (paper towel was clean - not black after this application) and then heated the empty pan on my electric stove for 15-20 minutes. The oil on the pan smoked a bit, some parts of the pan turned slightly ashy (grey). Mostly everything looked OK.   After I turned it off and cooled the pan, I now take a clean paper towel and rub the surface and I see black flakes! What is this? Is this the oil broken down? Can I cook in the pan or wash it with soap to eliminate the black flakes? Once washed and dried should I rub canola oil again? This is my first experience with cast iron. Any pointers will help. Book about mages who can harvest magical energy from the environment Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities.      Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices over $\mathbb R$ or $\mathbb C$. We say $A$ and $B$ are similar, or that $A$ is similar to $B$, if there exists a matrix $P$ such that $B = P^{-1}AP$. What is the best way to dump entire objects to a log in C#? Proving the geometric sum formula by induction $$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$  I want to prove this by induction. Here's what I have.  $$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$  I wanted to factor a $q^{n+1}$ out of the second expression but that 1- is screwing it up... Select an area in rep graph not working on Chrome The "select an area by dragging across the lower chart" thing is not working on Chrome.  Tested with Chrome 13.0.782.220 m, works with Firefox 6.0.1 and IE 8  Quirk: For some reason, if I open an incognito window, it works. Tried clearing the cache and cookies, still not working. How can I propose a new site? In Guardians of the Galaxy vol. 2, we see long-distance space travel. It involves piloting your ship into some sort of "jump point" that opens up in space, taking the shape of a honeycomb-like structure with a single hex missing.  The way it's describe, these points are fixed in space: early in the movie, the Guardians have to out-run an enemy fleet to reach the one they need; later on, we see them fly through a very long series of them in sequence. It's also made clear that doing too many of these "jumps" in a row has consequences for the passengers.  The process seems to be similar to, say, the Mass Effect relay setup, mixed with a bit of Star Wars hyperspace travel.  Is this the first time we have seen long-distance space travel in the MCU? Is there anything from the source material that explains how this works from a logistical standpoint? If there's any information on the "science" behind it, that would also be helpful.