I'm trying to see how    $$ [\mathbb Q(\alpha):\mathbb Q]=4,$$   where   $$ \alpha^5 = 1,\ \alpha \neq 1 .$$   I don't understand why it's 4 and not 5.  From my knowledge the basis could be   $$ {1, \sqrt[5]{1}, \sqrt[5]{1^2}, \sqrt[5]{1^3}, \sqrt[5]{1^4}} $$  Why isn't this the case?  Help would be much appreciated, thanks
To give some background, I am attempting to solve the problem from Artin's algebra textbook which was already asked in this link:  Now, I would like to prove the following two claims which are not obvious to me. If I can prove these, I will be able to show that $\zeta_5\notin \mathbb{Q(\zeta_7)}$ as needed. I suspect that both claims will have similar proofs.  Claim 1: $\zeta_5$ has degree $4$ over $\mathbb{Q}$ Claim 2: $\zeta_7$ has degree $6$ over $\mathbb{Q}$  Attempt to prove Claim 1 Let $α=\zeta_5$. I would like to find the monic irreducible polynomial for $α$, but unfortunately I do not know what it is yet. I see $α=cos(\frac {2π}5)+isin(\frac {2π}5)$ and $α^4=cos(\frac {4π}5)+isin(\frac {4π}5)$. Please help me to find the irreducible polynomial, and (ideally) to show that it is irreducible.
How to extract code of .apk file which is not working? How can I show that $ab \sim \gcd (a,b) {\operatorname{lcm} (a,b)}$ for any $a,b \in R \setminus \{0\}$? Let $R$ be a UFD. Then for any two elements $a,b \in R \setminus \{0\}$ $ab \sim \gcd (a,b){\operatorname{lcm} (a,b)}$.   My attempt :  Let $d = \gcd (a,b)$.Then $d|a$ and $d|b$.Then $\exists x,y \in R$ such that $a = dx$ and $b = dy$.Now it is to be shown that $c = dxy$ is the lcm of $a,b$.Which I find difficulty to show.It is clear that $a|c$ and $b|c$. Now if $u$ is any common multiple of $a$ and $b$.Then $dx|u$ and also since $y|b$ we have $y|u$ i.e. $dxy|u^2$ $\implies c|u^2$.But I have to show $c|u$ to complete the proof which I fail to prove.Please help me to complete the proof.  Thank you in advance. Blocking web sites with Windows Firewall Shouldn't this be a pretty straightforward operation? However, I see there's neither a size() nor length() method. I am really confused.  Which preposition is correct?     She is in/at the park.   They are in/at the park.   I am in/at the park.   Should I use in or at in these sentences? There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (given in the problem). There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (all the candies must be spread out) : (a) candies and boxes are different; (b) candies of the same diiferent boxes should not be empty cartons: (c) candies equally new, boxes are different; Edit :(d) candies are different, the boxes are the same, there should be no empty boxes; (e) candies of different boxes alike.(Edit : candies are different, boxes are equal) Specify the display type that matches the placement, if possible. My Answers : (a) Each layout is encoded with a word of $n$ letters from the alphabet of 't' letters $\implies$ possible $n^t$ variants. (b) Write, $n$ candies in the form of balls ina line, we need to put $(t-1)$ partition in $(n-1)$ place, but we can't put two partitions on one place, so we get : $^{t-1}C_{n-1}$. (c) First we need to choose a candy in a box (in the first box $n$ ways, in the second box $(n-1)$, $\cdots$ in the $t-th$ : $(n-t+1)$ ways $\implies$ total $n!/(n-t)!$ methods, and then we distribute the remaining $(n-t)$ candies into $t$ boxes, this is encoded with owrds from the alphabets (for each candy $t$ variants) $\implies$ $t^{n-t}$. The Answer is : $\frac {n!}{(n-t)!}*t^(n-t)$ Are my answers (a), (b) &amp; (c) are correct? For (d) &amp; (e) I don't know how to proceed? Please help me. Why doesn't Java allow to throw a checked exception from static initialization block? Rewrite rules goes away Let $ X \subset \mathbb{R}$ be a non-empty, open set and let $f: X \rightarrow \mathbb{R}$ be a continuous function. Show that the inverse image of an open set is open under f, i.e. show:  If $M \subset \mathbb{R}$ is open, then $f^{-1}(M)$ is open as well.  I think that this should basically follow from the definitions, but I'm still having some troubles with the proof.  Suppose $M \subset \mathbb{R}$ is open, then it follows that $\forall z \in M, \exists r &gt; 0: K_r(z) \subseteq M$  Since f is continuous it follows for each $x_0 \in X$ that for each $\epsilon &gt; 0 ,\exists \delta &gt; 0 : |f(x)-f(x_0)|&lt; \epsilon, \forall x \in X: |x-x_0|&lt;\delta$  Now I need to show that $f^{-1}(M)$ is open, so that for each $z \in M, \exists r&gt;0: K_r(f^{-1}(z_0)) \subseteq f^{-1}(M)$  I'm not sure how to prove this. I guess since $z = f(x)$ and since M is open we have that $\exists r&gt;0:\{z:|z-z_0|&lt;r\}\subseteq M$. And since f is continuous: $|f(x)-f(x_0)| = |z-z_0|&lt; \epsilon$ for all $|x-x_0|&lt;\delta$, but since $|x-x_0|=|f^{-1}(z)-f^{-1}(z_0)|&lt;\delta$ it somehow follows from that that $f^{-1}(M)$ is open?!  Which is just horrible and probably completely wrong. But I'm really confused from all the definitions right now and need some help please. How do I concatenate JavaScript files into one file? Is it correct to hyphenate with compound premodifiers? If so, where is the hyphen placed? Fastest way to remove non-numeric characters from a VARCHAR in SQL Server I'm writing an import utility that is using phone numbers as a unique key within the import.  I need to check that the phone number does not already exist in my DB. The problem is that   phone numbers in the DB could have things like dashes and parenthesis and possibly other things. I wrote a function to remove these things, the problem is that it is slow and with thousands of records in my DB and thousands of records to import at once, this process can be unacceptably slow. I've already made the phone number column an index.  I tried using the script from this post:   But that didn't speed it up any.  Is there a faster way to remove non-numeric characters? Something that can perform well when 10,000 to 100,000 records have to be compared.  Whatever is done needs to perform fast.  Update Given what people responded with, I think I'm going to have to clean the fields before I run the import utility.   To answer the question of what I'm writing the import utility in, it is a C# app. I'm comparing BIGINT to BIGINT now, with no need to alter DB data and I'm still taking a performance hit with a very small set of data (about 2000 records).   Could comparing BIGINT to BIGINT be slowing things down?  I've optimized the code side of my app as much as I can (removed regexes, removed unneccessary DB calls). Although I can't isolate SQL as the source of the problem anymore, I still feel like it is. When do I apply Significant figures in physics calculations? I'm a little confused as to when to use  for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.  Would I need to apply the rules of significant figures to this step of the problem? $$ C = 2\pi (1000~\mathrm{m}) = 6283.19 $$  Or do I just need to apply significant figures at this step? $$ \text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s} $$  Should I round $125.664~\mathrm{m}/\mathrm{s}$  to  $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two? What does --shell-escape do? I stumbled upon an example, which needed to be compiled with the additional option --shell-escape. Using this option it worked just fine, but I'm curious:     Why or when is this option needed and what does it actually do? I want to try to convert a string to a Guid, but I don't want to rely on catching exceptions (   for performance reasons - exceptions are expensive for usability reasons - the debugger pops up  for design reasons - the expected is not exceptional   In other words the code:  public static Boolean TryStrToGuid(String s, out Guid value) {     try     {         value = new Guid(s);         return true;     }     catch (FormatException)     {         value = Guid.Empty;         return false;     } }   is not suitable.  I would try using RegEx, but since the guid can be parenthesis wrapped, brace wrapped, none wrapped, makes it hard.   Additionally, I thought certain Guid values are invalid(?)    Update 1   had a good idea to catch only FormatException, rather than all. Changed the question's code sample to include suggestion.    Update 2  Why worry about thrown exceptions? Am I really expecting invalid GUIDs all that often?   The answer is yes. That is why I am using TryStrToGuid - I am expecting bad data.  Example 1 . I might be parsing folder names, checking to see if the text after the final . is a GUID.  c:\Program Files c:\Program Files.old c:\Users c:\Users.old c:\UserManager.{CE7F5AA5-6832-43FE-BAE1-80D14CD8F666} c:\Windows c:\Windows.old   Example 2 I might be running a heavily used web-server wants to check the validity of some posted back data. I don't want invalid data tying up resources 2-3 orders of magnitude higher than it needs to be.  Example 3 I might be parsing a search expression entered by a user.     If they enter GUID's I want to process them specially (such as specifically searching for that object, or highlight and format that specific search term in the response text.)    Update 3 - Performance benchmarks  Test converting 10,000 good Guids, and 10,000 bad Guids.  Catch FormatException:    10,000 good:     63,668 ticks    10,000 bad:   6,435,609 ticks  Regex Pre-Screen with try-catch:    10,000 good:    637,633 ticks    10,000 bad:     717,894 ticks  COM Interop CLSIDFromString    10,000 good:    126,120 ticks    10,000 bad:      23,134 ticks   p.s. I shouldn't have to justify a question. product distribution of two uniform distribution, what about 3 or more How can I port forward with iptables? Normal Subgroup Counterexample Im having trouble with the second part of this question,  Let $H$ be a normal subgroup of $G$ with $|G:H| = n$,  i) Prove $g^n \in H$ $\forall g \in G$ (which i have done)  ii) Give an example to show that this all false when $H$ is not normal in $G$.(which I am having trouble with showing)  Any suggestions? I am running the latest version of Linux Mint with Cinnamon. I'm trying to map Caps Lock to Ctrl, but I cannot figure out how to do it. All web searches I've done have led me to older versions of Linux Mint (there is no keyboard layout option in my settings). How can I do this? In an integral domain every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible.