I was wondering if there is some general method to find the Galois group of $p(x)=x^n-a$ over $\mathbb{Q}$. I know that the splitting field of $x^n-1$ over $\mathbb{Q}$ is $\mathbb{Q} (\zeta)$ with Galois group $\mathbb{Z}_n^{\times}$ where $\zeta$ are the primitive roots of unity, but what about $p(x)=x^n-a$ generally? Can I use this knowledge to find the Galois group of $p(x)=x^n-a$? I know that the solutions of $p(x)=x^n-a$ are of the form $a^{1/n} \zeta^i$ for $i=0,1,...n-1$. I am mostly interested in the case where $a^{1/n}$ is irrational.
I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q} $ Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having a nice "basis.  The splitting field is given by $\mathbb{Q}(\zeta_n,\alpha)$ , where $\zeta_n$ is a primitive root of unity , and $\alpha$ is some number such that $\alpha^n = a $. Well first of all, I want a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different (in this case the computation of the galois group is very simple).  For example one easy case, it's when $a&gt;0$, then $(a)^{\frac{1}{n}} \in \mathbb{R}$ , so clearly the minimal polynomial of $(a)^{\frac{1}{n}} , \zeta_n$ are distincs, and I'm done. If $n$ is odd then , it's also easy, since one root it's also real, for example $x^3-3 $, the real root is $ \root 3 \of { - 3}  =  - \root 3 \of 3  $ , so I can consider the splitting field as $\mathbb{Q}(-\root 3 \of 3 , \zeta_3 )=\mathbb{Q}(\root 3 \of 3 , \zeta_3 )$. The difficult case is when $n$ is even and $a&lt;0$ , for example  $x^8+20$ or $x^4+20$ in some cases as in the second, there are particular cases since there exist algorithms for the Galois group of quartics, but in general. It can be done?  $\underline{Remark}$ I'm searching a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different since in this case the computation of the galois group is very simple.
How can I quickly get a player back into a session after their character dies? As a GM, what kind of techniques could I use to help a player re-join the gaming session after their character dies? I want to avoid any major interruptions to the flow of the adventure. I would also like to avoid having to end the session immediately or making them sit on the side-lines for the rest of the session. The Lie algebra of $SL_n(\mathbb C)$ are the matrices where the trace is $0$. But what is the Lie algebra of $SL_n(\mathbb H)$ where $\mathbb H$ is the quaternions? This question exists as it fills a specific criterion.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will likely be closed per the . More information on the .   What kind of desktop environments or shells are there for Ubuntu users to install?  Please list a desktop environment or shell for each:   a description on why you like or suggest it (features, performance, etc.). a good screenshot, preferably of it running on Ubuntu and showing off some of its features. the minimal requirement for it to be usable, If there's any setting to lower it's requirement (Like kde's low-fat profile); how to enable the said setting some instructions on how to set it up if in the repositories, please provide a software centre link Given an integer $n$, we are asked to investigate about the existence of integer divisors of $n^2+1$ of the form $4k+3$. Can you provide some insights about it? I have a script that reads a text stream and generates a file of sed commands that is later run with sed -f. The generated sed commands are like:  s/cid:image002\.gif@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1922/g s/cid:image003\.gif@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1923/g s/cid:image004\.jpg@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1924/g   Assume the script which generates the sed commands is something like:  while read cid fileid do     cidpat="$(echo $cid | sed -e s/\\./\\\\./g)"     echo 's/'"$cidpat"'/https:\/\/mysite.com\/files\/'"$fileid"'/g' &gt;&gt; sedscr done   How can I improve the script to ensure all regex metacharacters in the cid string are escaped and interpolated properly? I am using an external application to enter data into a Google Spreadsheet.  This workbook is then referenced by a separate workbook (with the IMPORTRANGE(spreadsheet_key, range_string) function) that manipulates and interprets the data.  My problem is that whenever I personally modify the referenced workbook, the referencing workbook updates just fine, but whenever the external application modifies it, it does not get updated.  I have tried editing the "Spreadsheet settings" so that recalculation is done every minute and on every change.  Furthermore, I have also installed an extension in Google Chrome that auto-refreshes the page every hour.  However, the data does not get re-imported from the referenced workbook.  Even if I copy the formula into a new cell, the data is still not re-imported.  Is there anything that I can do to the referencing workbook to trigger Google Sheets to re-import the data?  Edit: Just to be clear, I currently have one workbook with data entered by an external application in it (call it the "source sheet"), and another workbook with an IMPORTRANGE function in it (call it the "referencing sheet").  The data shown by the IMPORTRANGE function in the "referencing sheet" does not include any data entered by the external application since I last personally edited the "source sheet".  Also, both workbooks are using the new Google Sheets.  Edit: Also, this question is not the same as  because I am using the function given as the solution to that question to import data from a spreadsheet.  The problem is not how to import the data, but rather how to update the source for the data.  My assumption would be that Google would have taken care of this for me, but the data in the "referencing sheet" does not get updated, and the only way I can find to get it to update is to physically go into the "source sheet" and edit it myself. I have a 2 year old son and another baby on the way. I just recently found out I'm pregnant but not sure how far just yet. My son rammed into my stomach like he is always doing but this time it made me start having small pains in my stomach. I rushed to the bathroom to make sure there wasn't any bleeding.  I'm concerned his rough play could hurt the fetus if it continues. How should I put a stop to it? How is Bohr's model related to electron cloud models via the correspondence principle? Going into a directory linked by a link When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? How to negate a specific symbol? I want to negate the symbol whose code is \models (this would be specific to Formal Logic). How can I do that? I tried a number of combination, but didn't find how this could be done. I tried \neqmodels, \nemodels, \notmodels, \nmodels  but all of them are wrong.  Is there a general way to negate symbols? I'm trying to solve a question where I need to write $\frac {1}{1+z^2} $ as a power series centered at $z_0=1$  I'm not allowed to use taylor expansion. So my first thought was to rewrite the function in a form where I can apply the basic identity:  $$ \frac{1}{1-x} = \sum_{n=0}^\infty x^n $$  So let's rewrite the original function $f(z) = \frac {1}{1+z^2} $ and use above identity:  $$\frac {1}{1+z^2} = \frac {1}{1-(-z^2)} $$ But observe that I need to center it at $z_0=1$ so I rewrote it to:  $$\frac {1}{1-(-z^2)}=\frac {1}{2z-(-(z-1)^2)} $$  I can factor out $\frac {1}{2z}$ and use the basic identity to get : $$\frac {1}{2z}\frac {1}{1-\frac{(z-1)^2}{2z}} = \frac {1}{2z}\sum_{n=0}^\infty \left(\frac{-(z-1)^2}{2z}\right)^n =\frac {1}{2z}\sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^n} = \sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^{n+1}}$$  The problem is if I ask wolfram alpha: power series$ \frac{1}{1+z^2} $centered at z = 1 it will give me a different answer that doesn't match the sum I found.  Can anyone help me figure this one out? What am I doing wrong? Let $T: V \rightarrow V$ be a linear operator.  I need to demonstrate that if all nonzero vectors of $V$ are eigenvectors of $T$, then there is one specific $\lambda \in K$ such that $T(v) = \lambda v$, for all $v \in V$.  I understand that, if all nonzero vectors of $V$ are eigenvectors of $T$, then $T$ must be a scaling transformation. It just stretch or shrinks vectors, but doesn't change their  directions.  So, the statement says that if it happens, then, there is a single $\lambda$ such that $T(v) = \lambda v$. In other words, if there is such transformation, then it scales all vectors by the same scalar $\lambda$.  Applying the transformation to our standard basis vectors, we have: $$ T(e_1) = \lambda_1 e_1 \\ T(e_2) = \lambda_2 e_2 \\ \vdots \\ T(e_n) = \lambda_n e_n $$  I understand I need to prove that $\lambda_1 = \lambda_2 = \dots = \lambda_n$, but I can't see how!  EDIT  $$ v = c_1e_1 + c_2e_2 + \dots + c_ne_n \\ T(v) = \mu v = \lambda_1c_1e_1 + \lambda_2c_2e_2 + \dots + \lambda_nc_ne_n \\ $$  Since what's multiplying $v$ coordinates is $\lambda_i$, then all of them must be $\mu$. I'm not sure how to 'mathematize' this. Is this idea correct?  EDIT 2 Extending the left hand side of EDIT 1, we have: $$ \mu v = \lambda_1c_1e_1 + \dots + \lambda_nc_ne_n \\ \mu(c_1e_1 + \dots + c_ne_n) = \lambda_1c_1e_1 + \dots + \lambda_nc_ne_n \\ \mu c_1e_1 + \dots + \mu c_ne_n = \lambda_1c_1e_1 + \dots + \lambda_nc_ne_n \\ $$  And since $e_i$ are linearly independent, $\mu = \lambda_1 = \lambda_2 = \dots = \lambda_n$. Is this proof correct? How can I add "Show desktop" to the GNOME dash or Ubuntu Dock? Linux: Case-INSENSITIVE Filesystem What is the formal definition of polynomial ring of several variables? Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based on first-order logic, one can only talk about something that actually exists (as a set). So as its name says, indeterminate variable is not a sentence in $ZFC$. Nevertheless, we can formally define a polynomial ring $R[X]$ as a subset of $R^\omega$ whose support is finite. ($R$ is a commutative ring with unity) Just like a single variable, I want to know what would be the formal definition of a polynomial ring of several variables. What would be a formal definition? === I have figured out a candidate. That is,  Candidate for a &quot;Definition of a polynomial ring $R[X_1,...,X_n]$&quot;. ($R$ is a commutative ring with unity) Let $R[X_1,...,X_n]$ be the set of all functions $f:\omega^n \rightarrow R$ whose support is finite. Define $(f+g)(a,b,c)=f(a,b,c)+g(a,b,c)$ and $(f•g)(a'',b'',c'')=\sum_{a+a'=a''} f(a,b,c)•g(a',b',c')$ (Note that in this way, $+,•$ are well defined.) Then, it can be checked that $(R[X_1,...,X_n],+,•)$ is a ring.  Is this a formal definition of a polynomial ring? Why is the comma splice an error? It's always drilled into us that we shouldn't use comma splices. They're bad, to be avoided and poor form.  But why? What makes the comma a faulty choice in something like this:     I never liked school, it was boring and repetitive most of the time.   It could be replaced with a semicolon, dash, period or colon and be "correct." Yet to a fair amount of people, the comma seems fine. If it weren't, it wouldn't be found so often in online forums, text messages, emails and so on.  What makes the comma between two sentences an error? How to make shortcuts on Ubuntu 17.10? R/mgcv: Why do te() and ti() tensor products produce different surfaces? Which facility upgrade unlocks maneuver nodes in KSP What is the name for moments, a parameter that is calculated