Let $\zeta_{21}$ be a primitive 21st root of 1. Describe explicitly the elements of the Galois group $\text{Gal}(\mathbb Q(\zeta_{21}):\mathbb Q)$ and determine the structure of this group.   From finding the minimal polynomial I know that the order of the Galois group must be 12. So $\text{Gal}(\mathbb Q(\zeta_{21}):\mathbb Q)≅(\mathbb Z/21\mathbb Z)^{\times}$.  I have seen online that $(\mathbb Z/21\mathbb Z)^{\times}$ has the structure $C_2 \times C_6$, how has this been found?   How can I describe the elements of this Galois group?
$\zeta_{15}$ is $15$th primitive $n$th root of unity.   Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$  I know that if $p$ is prime then $G=Gal(\mathbb{Q}(\zeta_{p})/\mathbb{Q})=\mathbb{Z_{p}}^*$ but when $p$ is not prime, I am not show how to solve this if not prime.  Does my required group have any relation to some cyclic group $\mathbb{Z_n}$, i.e. abelian  Would really appreciate your guidance as I have not got my head around these concepts of cyclotomic fields and Galois structure. Thanks
Counting the number of surjections. How many functions from set $\{1,2,3,\ldots,n\}$ to $\{A,B,C\}$ are surjections? $n \geq 3$  Attempt  I was hoping to count the number of surjections by treating $A,B,C$ like bins, and counting the number of ways to fill them with $1,2,3,\ldots,n$ such that no bins are empty. This is equivalent to putting two "separator" bars in between the $n$ numbers or $n-1 \choose 2$.  However, I think I am missing a step here: prior to putting in the separator bars, I should permutate the $n$ elements. So, multiply by $n!$.  The order of elements in the same bins do not matter. So I should be divided by the number of permutations of the elements in each bins. This is where I am stuck.  Additional Info  I made the question up myself. Feel free to modify the question into a more solvable form if it is too difficult/complicated. Continuity of absolute value of a function Let $f(x)$ be a continuous function. Prove that $\left|f(x)\right|$ is also continuous.  Is it correct to say that, by the reverse triangle inequality, $\left|f(x)-f(c)\right| \geq \left|f(x)\right|-\left|f(c)\right|$ in all cases, so we will always have that for all $\left|x-c\right|&lt;\delta$ implies $\left|f(x)-f(c)\right|\leq \epsilon$?  I am not sure if my solution adequately uses the functional form, since I just used the equation and not my actual function. Please help with the proper solution! What is the purpose of .bashrc and how does it work? How do I copy vertex groups from one object to another? So, I decided that a previous project contained exactly what I needed to expedite a new project, so I appended the appropriate objects form the previous project to the new one. This was great, because I didn't have to define new materials, &amp;c.   Regrettably, though, I found out that when it came to vertex groups, where it would have been as helpful to be able to copy vertex groups from one object to another as it was to copy materials, I couldn't figure out how to do so.   Am I looking for a feature which does not yet exist?  NOTE: I did not expect the vertex groups to have assignments in the target objects, but it would have been handy not to have had to retype 25 names. $\displaystyle \int\nolimits_0^8 \sqrt{x^4+4x^2}\,dx$.   Alright, so I thought I had this figured out. Here's what I did:   I factor out an $x^2$ to get $\sqrt{x^2(x^2+4)}$. I let $x = 2\tan(\theta)$, therefore the integrand is $\sqrt{4\tan^2(\theta)  (4\tan^2(\theta) + 4)}$. Factor out a 4 and it becomes $\sqrt{(16\tan^2(\theta)  (\tan^2(\theta) + 1))}$ Which equals $\sqrt{16\tan^2(\theta) \sec^2(\theta)}$ This is easy to take the sqrt of. The integrand becomes $4\tan(\theta)\sec(\theta)$. Now, the integral of this is $4\sec(\theta)$ And it's evaluated from $0$ to $\arctan(4)$ right? Because as $x$ goes to $0$, so does $\theta$, and as $x$ goes to $8$, $\theta$ goes to $\arctan(4)$... But the end result $(4 (\sec(\arctan(4)) - 1) )$ isn't the correct answer   I put it into WolframAlpha and I get $(8/3) (17\sqrt{17} - 1)$, which is the right answer. How did they get that? (there's no "show steps" option)  Any help is greatly appreciated!  PS, what's the syntax for doing sqrts and exponentials? What are Click packages? I noticed that Ubuntu SDK recently had some updates and it installed a program named "Click". I would like to know what they are and how to use them.  Also, would they make Debian packaging easier? How many files can a windows folder contain?  This question is asked  but then the question text pollutes the question with issues of "rules of thumb"  So to be clear, this question is: How many files can a windows folder contain? While using Wild Shape, do I use Unarmored Defense or the creature's Armor Class? Is there a way to avoid SSH typing delay? lists Ubuntu 4.10 (Warty Warthog) as the first Ubuntu released by Canonical Ltd.  But I wanted to know whether there ever was an Ubuntu 1.x, as in Windows or Mac OS. If yes, is it available somewhere? Seeking English term for relationship between two members of extended family? Here at the Genealogy and Family History Stack Exchange we see many questions about relationship mapping which is also known as:   relationship calculation kinship terminology etc   If you are just trying to understand how to calculate "cousins and removes" then  provides an excellent guide.  However, if you are seeking an English term for a more complicated relationship between two members of an extended family, rather than asking a question like:          site users are encouraged to first review the answers below.    Note: This Q&amp;A arises from  and is set to  to help make it canonical. I'm looking for a term for audio in form of the word visual.  Visual is defined as      of or relating to the sense of sight   What could you call the sense of hearing? Also, what do you call this form of a word? (i.e verb, noun...)  It should fit the x in the following sentence:     Is it possible to fall in love with someone without visual or x contact?   I'm not a native English speaker, so I couldn't find the terms for searching this. Why does the set of all singleton sets not exist? How can I get free or trial ExactTarget instance Is there a word for the sadness over "What might have been"? Context: You made the decision not to see a person any more. The relationship was good, you were both happy with one another but the one main stumbling block (for you) in the relationship was never dealt with. It remained, a permanent fixture, that forced you to make that sad, and final decision.   Is there a word, an idiom, or a famous line that expresses the sadness for something that might have been; e.g. a rich and fulfilling relationship. It's not nostalgia, because that expresses a wistfulness for the past.   Instead, you are wistful for the future. How does substituting Butter for Margarine/Shortening affect the Recipe I know that Margarine / Shortening are usually softer than butter, but what other differences are there? Can I just melt the butter and call it good or will this effect my final Product? I have seen Margarine / Shortening a lot in cookies, but I am looking for a general answer. What is the word for when someone is overly nice and actually isn't nice at all? What is the word for when someone is overly nice and actually isn't nice at all because of the unnecessary added information/comments? For example, they will say things like "Your hair looks so good that wayyyy omg you should do it that way ALL the time! Looks way better than how you usually do it." And the word I'm looking for is not passive aggressive and it is not sarcasm. There is a word for these types of people. What is it? These people think they're being kind but actually blunt and rude. Another example would be for someone to say "hey you're beautiful, but maybe if you didn't wear harsh makeup you would look better" and they intend to sound like they're being nice when you know they don't like you and are saying in out of spite. Prevent execution of parent event handler How to generate a random integer number from within a range How to control Windows 7 snap feature with two monitors? There is an  feature in Windows 7. However, it seems to not work when dragging a window to the edge that is adjoined to the second monitor. How to make it work properly? Maybe some regedit modification or something?