Prove that $\sqrt[7]{2}\not\in\mathbb{Q}(\sqrt[7]{3})$ I need to show that $\sqrt[7]{2}\not\in\mathbb{Q}(\sqrt[7]{3})$ but I have no idea how to do it. I've tried to reason with the extension degrees but I haven't got any satisfactory result. Could someone help me? Thanks in advance.
Show the two fields are not isomorphic Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$.  By Eisenstein we know the two polynomials are irreducible and so they are minimal polynomials and both have degree $p$.
Can you use twinned spell on the same target? So me and my DM have been arguing on how Twinned spell works. Twinned spell:  When you cast a spell that targets only one creature and  doesn't have a range of self, you can spend a number of  sorcery points equal to the spell's level to target a second  creature in range with the same spell (1 sorcery point if  the spell is a cantrip).  I argued that &quot;a second creature&quot; can be a the same target as it is not worded as &quot;another creature&quot; For this example counter spell with twinned spell.  A wizard casts fireball on me I use CounterSpell#1 with Twinned Spell Wizard casts CounterSpell on my CounterSpell#1 I use him as the same target for my twinned Spell CounterSpell#2  Result countered Is this possible? how do you make clothes move along with your model? Do you rig the model first and then parent it to the body of the character then click cloth simulation? I am a little bit confused with the process. Thanks for the help. How to convert a string of bytes into an int? How can I convert a string of bytes into an int in python?   Say like this: 'y\xcc\xa6\xbb'  I came up with a clever/stupid way of doing it:  sum(ord(c) &lt;&lt; (i * 8) for i, c in enumerate('y\xcc\xa6\xbb'[::-1]))   I know there has to be something builtin or in the standard library that does this more simply...  This is different from  for which you can use int(xxx, 16), but instead I want to convert a string of actual byte values.  UPDATE:  I kind of like James' answer a little better because it doesn't require importing another module, but Greg's method is faster:  &gt;&gt;&gt; from timeit import Timer &gt;&gt;&gt; Timer('struct.unpack("&lt;L", "y\xcc\xa6\xbb")[0]', 'import struct').timeit() 0.36242198944091797 &gt;&gt;&gt; Timer("int('y\xcc\xa6\xbb'.encode('hex'), 16)").timeit() 1.1432669162750244   My hacky method:  &gt;&gt;&gt; Timer("sum(ord(c) &lt;&lt; (i * 8) for i, c in enumerate('y\xcc\xa6\xbb'[::-1]))").timeit() 2.8819329738616943   FURTHER UPDATE:  Someone asked in comments what's the problem with importing another module. Well, importing a module isn't necessarily cheap, take a look:  &gt;&gt;&gt; Timer("""import struct\nstruct.unpack("&gt;L", "y\xcc\xa6\xbb")[0]""").timeit() 0.98822188377380371   Including the cost of importing the module negates almost all of the advantage that this method has. I believe that this will only include the expense of importing it once for the entire benchmark run; look what happens when I force it to reload every time:  &gt;&gt;&gt; Timer("""reload(struct)\nstruct.unpack("&gt;L", "y\xcc\xa6\xbb")[0]""", 'import struct').timeit() 68.474128007888794   Needless to say, if you're doing a lot of executions of this method per one import than this becomes proportionally less of an issue. It's also probably i/o cost rather than cpu so it may depend on the capacity and load characteristics of the particular machine. Multinomial logistic regression analysis or binary logistic regression derivative of a determinant of matrix Let $A,B\in M_{n\times n}(\mathbb{R})$, prove that $$(\frac{d}{dt}\det(A+tB))_{t=0} = Tr(cof(A)^{T}\cdot B), $$ where $cof(A)$ is the cofactor matrix of $A$. I tried expanding it using row expansion, but it gets messed up. Any help? What is the reason the log transformation is used with right-skewed distributions? I once heard that      log transformation is the most popular one for right-skewed distributions in linear regression or quantile regression   I would like to know is there any reason underlying this statement? Why is the log transformation suitable for a right-skewed distribution?   How about a left-skewed distribution? What is the recommended CIDR when creating VPC on AWS? I have been creating AWS VPCs and I am wondering if there is a recommended CIDR value when creating VPCs. What are the factors that I must consider when choosing a CIDR and does the CIDR value affect the performance of the network? I have an Android device which is running Android 4.0.3 (or something like that).  My little brother decided to try and guess my lock screen pattern the other day, and ended up locking me out.   I have since been unable to log back in to the tablet using my Google Account details. I provided the correct login details, however it says they are incorrect. Could this be connected to the fact I have no internet connection?  I have no internet connection as the device is WiFi only, and the WiFi is currently turned off. I cannot turn it on, as I am locked out.  How can I log back into my tablet without wiping the user data? Where are the Unity desktop wallpapers located? What is the first reference to an internet of computers in science fiction? Recovery of deleted folders in home folder By accident I've removed some folders (they were empty) in my home folder. I've created a new ones but they doesn't have proper icons etc... how can I recreate them step by step? I'm looking to generate a set of 5 random variables and enforce a dependence structure between them and onto a dependent variable $Y$.  I understand how to generate correlated random variables for multivariate normal, but not when mixing different types.  Below is a little more than I need, but I'm hoping someone can give me a general way of solving this problem...   $X_1$ and $X_2$ need to be highly correlated Bernoulli variables.  $X_3$ needs to take one of 5 categorical values, call them "A"..."E".  $X_4$ needs to be normal, and negatively correlated with $X_1$, $X_2$.  $X_5$ needs to approximate test scores from $0$ to $100$ with a high skew, so gamma probably.  $X_5$ needs to be positively correlated with $X_1$, $X_2$, $X_4$.   Each of these variables must impact a "success/occurrence" Bernoulli distributed variable $Y$.  How would I begin?  I would like to enforce correlation both between the values of $X$, and also between each $X$ and $Y$.  (The categorical correlations seem particularly confusing to me.) Determine if $.ajax error is a timeout Multivariable limit proof: $\lim\limits_{(x,y)\rightarrow (0,0)}\frac{\left|x\right|^a\left|y\right|^b}{\left|x\right|^c + \left|y\right|^d} = 0$ When I take the pth root, the bottom of the p comes very close to the \sqrt symbol.  In the New Century Schoolbook font (which I am using), the p actually intersects the \sqrt symbol, which looks even more terrible.  Is there a clean way to fix this?  \documentclass{article} \usepackage{fouriernc} % use the New Century Schoolbook font \begin{document} When I write $a^{(1/p)}$ as $\sqrt[p]{a}$, the bottom of the $p$ touches the top of the root symbol, which looks ugly. Even when the root symbol is larger, such as with $\sqrt[p]{\frac{1}{n}}$, it still looks bad (since the tail of the p almost hits that line). \end{document} To evaluate limit of sequence $\left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$ How do I evaluate the limit of the following sequence $$a_n  = \left(\left( 1 + \frac1n \right) \left( 1 + \frac2n \right)\cdots\left( 1 + \frac nn \right) \right)^{1/n}$$   I tried to take log and then using cauchy $1^{st}$ theorem of limits but I coudn't do it.  Could someone kindly help? "Prometheus" is meant to be a prequel to "Alien", happening 30 years before ().  If that's the case, how come one of the very earliest spaceships, Prometheus from 2089, has a spiffy new technology, while Nostromo has antique text based CRTs, mechanical IBM style keyboards, and all the associated technologies screaming "1980s called, and they want their IBM PC back!" ?  The great visuals of this can be seen in this Youtube Video  (ignore the russian voice-over - the guy is a movie critic who is complaining pretty much of the same discrepancy listed above).  Please note that I'm looking ONLY for in-universe answers (movies, scripts, explanations from creative team). I'm utterly un-interested in the obvious out of universe "well, for 1979 Alien film, 1980s technologies was the future, and 2012 movie is expected to look futuristic in 2012, duh" or speculations that make sense but aren't confirmed in canon.          And for those who don't believe me... to quote the immortal quip from Men in Black 2:   Old and busted:        New hotness: Why isn't .profile sourced when opening a terminal? Just installed Ubuntu 14.04.1 LTS.  According to .profile,  # ~/.profile: executed by the command interpreter for login shells.  # This file is not read by bash, if ~/.bash_profile or ~/.bash_login exists.   There is no ~/.bash_profile or ~/.bash_login.  It does not seem to source .profile when opening a terminal.   Why doesn't it source .profile when opening a terminal? Why is .profile there if it is not sourced? How do I know/control what is sourced when a new terminal is opened? When a new terminal is opened, does it do a "login"? Where is a good place to add $HOME/bin to my path? .bashrc? Will that cause a problem because .profile has the same code? I am planning an hour and a half ride with my 1 month old baby girl. I heard that being in the car seat for a long time is not good for newborns.  Is an hour and a half too long? Should we do a stop in the middle of the trip? TP-LINK TL-WN823N Unable to connect to network