I'm trying to find the orbits of the Frobenius homomorphism $\phi_p: a \to a^p$ on $\mathbb{F}_{p^4} / \mathbb{F}_p$. I can see there are $p$ 1 orbits corresponding to the action on $\mathbb{F}_p$ but I think there are also some order 2 and 4 orbits but I'm having trouble finding how many of each there are.  This is part of an argument on trying to show how many irreducible polynomials of degree 4 there are over $\mathbb{F}_p$ because I think this number should correspond to how many order 4 orbits there are, as then the polynomial with roots in that orbit should be irreducible.  Thanks
I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$.     I am interested in counting how many such $p(x)$ there exist (that is, given $n\in\mathbb{N}$, $n\ge 1$, how many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$).   I don't have a counting strategy and I don't expect a closed formula, but maybe we can find something like "there exist $X$ irreducible polynomials of degree $n$ where $X$ is the number of...".  What are your thoughts ?
Commutator of two normal subgroups with trivial intersections So I tried looking everywhere but I'm really lost with this proof. I need to to show that:  Let $H$, $K$ be normal subgroups of a group $G$, satisfying $H\cap K = \{e\}$. Let $h\in H$, $k \in K$. (a) Prove that the commutator $w = hkh^{-1}k^{−1}$ satisfies $w \in H$ and $w \in K$. Then conclude that $w = e$ (b) Prove that $hk = kh$, for all $h \in H$, $k \in K$. Physics history book with some math I am looking for a book explaining physics from, say, Galileo and Newton till now; the book should be written using some math, similarly to my physics books when I was at the university (Halliday Resnik). I am searching a book that take care of explaining the true "evolution" of reasoning, discoveries, experiment to prove or disprove something.  I do not like a book that gives definitions without explaining what had happened. I also like to find, on that book, explanations on how actual measurements are taken (how can we measure the speed of light, the mass of celestial bodies, the attributes of electron, and how were the measurements taken in the past - how was possible to measure the speed of light in 1700-1800?).  I am not very interested in recent theories like strings, branes.... so it is OK if these are not discussed.  I like to find a lot of discussions on concept that are related, like for instance the implications on space dimensionalities, ether non-existence, strange facts, missing explanations. mail -s "subject" xxxxx@gmail.com &lt;test.html works, but only for plain text email.  What is the correct way to send HTML email using the Linux command mail? Design simulates perfectly but won't work on FPGA Earth moves how much under my feet when I jump? If I'm standing at the equator, jump, and land 1 second later, the  Earth does NOT move 1000mph (or .28 miles per second) relative to me,  since my velocity while jumping is also 1000mph.   However, the Earth is moving in a circle (albeit a very large one),  while I, while jumping, am moving in a straight line.   How much do I move relative to my starting point because of this? I  realize it will be a miniscule amount, and not noticeable in practise,  but I'd be interested in the theoretical answer. Bash/Korn shell script edited on Windows throws error '...^M: not found' I wrote a Bash script in Windows using Notepad++.  cxStopAllServicesOnSERVER.sh  #!/usr/bin/bash cd "/some/path" echo "Hi. Logs can be found at "`pwd`"/cxStartStopLogger.log" echo "["`date`"]*** STOPPING ALL SERVICES ON SERVER ***" &gt;&gt; "cxStartStopLogger.log" exit   Now after uploading it and setting the required file permissions, I tried executing it as follows:  bash-3.00$ cat cxStopAllServicesOnSERVER.sh #Let's have a look at the code. #!/usr/bin/bash cd "/some/path/" echo "Hi. Logs can be found at "`pwd`"/cxStartStopLogger.log" echo "["`date`"]*** STOPPING ALL SERVICES ON SERVER ***" &gt;&gt; "cxStartStopLogger.log"  bash-3.00$ # Code and hashbang 'looks' correct,  bash-3.00$ # if there is any issue with the format (EOL characters and such)  bash-3.00$ # we cannot see it using cat. bash-3.00$  bash-3.00$ sh cxStopAllServicesOnSERVER.sh cxStopAllServicesOnSERVER.sh[2]: /some/path^M:  not found Hi. Logs can be found at /some/path/cxStartStopLogger.log bash-3.00$ # Note that ^M appears at the end of the line.  bash-3.00$ bash -x cxStopAllServicesOnSERVER.sh + cd $'/some/path\r' : No such file or directory1.sh: line 2: cd: /some/path ++ pwd ' echo 'Hi. Logs can be found at /some/path/cxStartStopLogger.log Hi. Logs can be found at /some/path/cxStartStopLogger.log ++ date + echo '[Sun' Nov 18 00:28:17 EST '2012]*** STOPPING ALL SERVICES ON SERVER ***' bash-3.00$ # Note that '\r' return character appears at the end of the line.   Problem: When I change the code to korn shell I face the same issue. Appears that an incorrect character is getting added at the end of the line.   NOTE: I found the solution and have posted the same as an answer. Feel free to update or improve it to help other beginners that may face the same issue.  Thanks! How to set Oracle's Java as the default Java in Ubuntu? I recently installed Ubuntu 14.04, then I installed lamp-server and placed my webpages in /var/www directory, but when I opened localhost in the browser there was nothing. I think that happened because Zend updated Apache.  Whatever the reason may be, I want to know where to keep my php files so that I can access them from my browser. Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$ I want to solve this system of equations, I have been out of Maths for a long!!  $$\sqrt{x}+y=7$$  $$\sqrt{y}+x=11$$  Just wondering easiest step to find values for $x$ and $y$ from the above equations? Shell script- To fetch the content from the specific tag at all occurrence using shell scripting I have an XML file on my server. In that file, I have many tags in which there is a tag for example, "container" tag. I need to fetch the content of all the container tags content separately. Many thanks in advance. Handling of special LaTeX characters in text I have a list of tuples like this:  [ ('a', 1), ('a', 2), ('a', 3), ('b', 1), ('b', 2), ('c', 1), ]   I want to iterate through this keying by the first item, so, for example, I could print something like this:  a 1 2 3 b 1 2 c 1   How would I go about doing this without keeping an item to track whether the first item is the same as I loop around the tuples? This feels rather messy (plus I have to sort the list to start with)... Can I update an Mac App purchased by a different Apple ID? Need help identifying the fantasy book about a portal to a fantasy land in the basement of a house I can't remember any names of characters or locations where the story took place but I do remember the basic plot of this early 90's novel; a human man inherits his father's house. In the basement there is a doorway to a fantastic land where adventure ensues. He fights elves and wins their deadly elf swords which he buries under concrete at the end of the novel (because if you don't get rid of them they will come back and get you).  At the end of the novel he must remake the world in his own imagination, turning the tough Elf King into a cobbler and making Jesse Jackson President. I have been looking high and low and have drawn a blank.  Any help would be appreciated. Thanks! Continuous footnote numbering I would like footnote numbering to continue over pages and chapters. In my document it's reset every chapter. I'm using the report documentclass. What do I do if I don't have a META-INF folder or minecraft.jar? I have looked all over the Internet for solutions but I have got nothing.  I want to install the minepainter mod but I need a META-INF folder and a minecraft.jar. I just know that ls -t and ls -f give different sorting of files and subdirectories under a directory.   What are the differences between timestamp, modification time, and created time of a file? How to get and change these kinds of information by commands? In terms of what kind of information do people say a file is "newer" than the other? What kinds of information's change will not make the file different?   For example, I saw someone wrote:     By default, the rsync program only looks to see if the files are different in size and timestamp. It doesn't care which file is newer, if it is different, it gets overwritten. You can pass the '--update' flag to rsync which will cause it to skip files on the destination if they are newer than the file on the source, but only so long as they are the same type of file. What this means is that if, for example, the source file is a regular file and the destination is a symlink, the destination file will be overwritten, regardless of timestamp.   On a side note, does the file type here mean only regular file and simlink, not the type such as pdf, jpg, htm, txt etc? How to highlight paragraph (linebreak preserving) as well as itemize, tabular etc.? I want to highlight changes which are introduced by myself in the pdf document. So far the soul package is very neat but it does throw errors when I want to highlight not only paragraphs but also some itemize environments, and I don't want to highlight the items separately. Also the \cite{...} command is not supported by \hl. I recently posted a slightly different question:   BTW: What I liked most over other approaches like pdfcomments etc. is that the \hl command from soul has a very exact and clear output, no icons, no clicking etc. involved. Together with todonotes it's very good and almost all I need. jsp:useBean scope I've recently installed Ubuntu on Oracle's Virtualbox.  Now I want to install the guest additions.  I started up my Ubuntu guest OS, clicked "Devices" and chose "Install Guest Additions". Nothing happened, and I don't see the CD icon on my desktop.  How can I make this work?