The usage of "actually" in comparison
"Actually" vs "really"
Let $k$ be a finite field of order $q$ in characteristic $p$, let $n$ be a positive integer not divisible by $p$, and let $K$ be the splitting field of $X^n-1$ over $k$. Prove that $[K:k]$ equals the smallest positive integer $d$ such that $n\mid q^d-1$.  My approach: It's easy to show that if $k$ is a finite field of order $q$ with $\text{char} \ k=p$ then $k=\mathbb{F}_{q}$ with $q=p^m$ with $m\geq 1$. If $K$ is the splitting field of $X^n-1$ over $k$ then $K=\mathbb{F}_{q^r}$ for some suitable $r\geq 1$.  Let $\alpha \in K=\mathbb{F}_{q^r}$ be a root of $X^n-1$ then $\alpha^n-1=0$ and $\alpha\neq 0$.   Thus, $\alpha^n=1$ and $\alpha \in \mathbb{F}^{\times}_{q^r}$ where $\mathbb{F}^{\times}_{q^r}$ is the cyclic group of order $q^r-1$.  But I have some difficulties:  1) From $\alpha^n=1$ it follows that $o(\alpha)\mid n$. How to show that $o(\alpha)=n$? I was trying to show it using that $(n,p)=1$ but I failed.  Remark: Because if we can show that $o(\alpha)=n$ then the desired result follows immediately from cyclicity of multiplicative group of field $\mathbb{F}_{q^r}$.  I have spent one day in order to come up with this approach. So please do not duplicate this question and can anyone show how to answer my question, please.  Detailed explanation would be great! How to prove that you have significant ties with family in your home country? My understanding is that heat is essentially atomic vibrations. If there are almost no atoms, how can there be residual heat?  Also, as I understand in space there is no heat transfer via convection and it is mainly through EM radiation. In this case, why is it that if i were to go into space with no suit and just an airtight helmet through which I could breathe, I would still freeze to death very quickly? HP PSC 1410 all scanner's images are black I have a HP PSC 1410 printer with scanner inside.  I've been scanning documents and images with it for the last couple of years.  Recently my scanner has stopped performing as excepted and i can't scan or copy images\documents.  My problem is that every images\document i'm trying to scan - the result from the printer is a black page.  I've been googling about this issue and couldn't find any helpful stuff beside replacing the whole printer - which is something i'm trying to avoid.  Any ideas ? I've read something about the shipping lock of the scanner but couldn't find any information about how to check it.. How to make a meta box field a requirement How does the double strike mechanic interact with other mechanics like lifelink or trample? It is known that all $B$, $C$ and $D$ are $3 \times 3$ matrices. And the eigenvalues of $B$ are $1, 2, 3$; $C$ are $4, 5, 6$; and $D$ are $7, 8, 9$. What are the eigenvalues of the $6 \times 6$ matrix  $$\begin{pmatrix} B &amp; C\\0 &amp; D \end{pmatrix}$$ where $0$ is the $3 \times 3$ matrix whose entries are all $0$.  From my intuition, I think the eigenvalues of the new $6 \times 6$ matrix are the eigenvalues of $B$ and $D$. But how can I show that? I would like to know if there is a word or phrase for a systematic campaign to annihilate every descendant of a particular ancestor.   We have the word 'genocide' for killing off an entire race, and plenty of words for killing people based on how they are related to you (fratricide, filicide, et. al.), but looking over the  and the , I couldn't find a term for systematically killing an entire bloodline.   My question is prompted by a US news story about the horrific murder of 8 members of the same family, but those murders are not an example of what I'm thinking of because the children of the victims were left alive, so the motive for the killings was not to kill off the bloodline. The sense that some group had coordinated an attack against just one family is what made me wonder what the "not an entire race, just one bloodline" version of 'genocide' would be called.   Searching the Internet for "kill off" and bloodline sends me straight into the land of vampires, Illuminati, reptilian overlords, and, naturally, Priory of Sion nonsense so I haven't had much luck finding a thread to pull.   It may be that there is no word for it in the sense of murdering a human bloodline, because we just don't attempt kill off our royals down to the last heir often enough to coin a term for it. Maybe there is a scientific term for culling a line from a test population as part of biological research? I used python add in to make a tool bar in Arcmap that edits features on a map. It works on a file geodatabase, but not on an ArcSDE version one. I've been getting an FDO error: 0. The solutions that I've found all involve opening a workspace, but I'm getting an error doing that as well. Here is the relevant portion of my code:          outFolderPath = r"U:\tempStuff"         outName = "Stuff.sde"         databasePlatform = "SQL_Server"         instance = "SQL Server"         arcpy.CreateDatabaseConnection_management(outFolderPath, outName,                                                   databasePlatform,                                                   instance)         print "Centroid of buidling:(%r,%r) Side length: %r" % (xCoordinate,                                                                 yCoordinate,                                                                 bob)         print "workspace is %r" % outName #outName is workspace          #Start edit session with an undo/redo stack then start         #edit operation         print "Starting editing"         edit = arcpy.da.Editor(outName) #outName is workspace Line 86         edit.startEditing(True, True)         edit.startOperation()          #stuff happens          cursor = arcpy.da.InsertCursor( fc, ["SHAPE@"])         cursor.insertRow([stuff])             #stop the edit operation and then save changes         edit.stopOperation()         edit.stopEditing(True)          del cursor         print "Building should be drawn"   This is the exact error I've been getting:  Traceback (most recent call last):   File "C:\Users\153289\AppData\Local\ESRI\Desktop10.3\AssemblyCache\{0BEF4E61-6332-4BCA-A908-959CA3C0B4E7}\leetScripts_addin.py", line 86, in onClick     edit = arcpy.da.Editor(outName)#outName is workspace RuntimeError: cannot open workspace   What's am I doing wrong in my code? I'm trying to show that  $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$  Where $a_{i}$ are the roots of the function $f(x)$. I've tried to proceed by using a dummy function $g(x)$ and carrying out:  $$\int_{-\infty}^{\infty}dx\,\delta\big(f(x)\big)g(x)$$  Then making the coordinate substitution $u$ = $f(x)$ and integrating over $u$. This seems to be on the right track, but I'm unsure where the absolute value comes in in the denominator, and also why it becomes a sum.  $$\int_{-\infty}^{\infty}\frac{du}{\frac{df}{dx}}\delta(u)g\big(f^{-1}(u)\big) = \frac{g\big(f^{-1}(0)\big)}{\frac{df}{dx}\big(f^{-1}(0)\big)}$$  Can any one shed some light? Wikipedia just states the formula and doesn't actually show where it comes from. Why an R-module is an R/I-module precisely when it is annihilated by I? Let $I$ be an ideal of $R$ and let $M$ be an $R$-module. Prove that the formula $(r + I) \cdot m = r \cdot m$ for $r + I \in R/I$, and $m \in M$ makes $M$ into an $R/I$-module if and only if $i \cdot m = 0$ for all $i \in I, m \in M$.   I don't quite understand why we must have $i \cdot m = 0$ instead of $I \cdot m \in M$. This is for proving the only if part. AMT stock option scenario In a bash shell script, writing a for loop that iterates over string values R Difference between rbinom and sample I'm trying to model a Roulette game where if you bet on red (out of red(18/37), black(18/37), green(1/37)) and if you the ball lands on red you win 1 dollar and if it doesn't you lose $1. So in this case, is it a binomial distribution? Since there are only 2 outcomes from the gamblers perspective, red or not red. If thats the case are these 2 R functions the same?  set.seed(3) sample(c(0,1),100,replace=T,prob=c(19/37,18/37))     rbinom(n = 100, size = 1, prob = 18/37) I would like to install  chat platform on my Ubuntu 12.04 LTS system. Please, could anyone provide me the steps to successfully install this software?. I've been trying to install this software using wine but I couldn't succeed it. I recently installed the apache2 package using the Software Centre, I want to create an alias apacheres which will replace the more cumbersome command /etc/init.d/apache2 restart.  I opened the .bashrc file in my home directory, and it contains the following lines:  # Alias definitions. # You may want to put all your additions into a separate file like # ~/.bash_aliases, instead of adding them here directly. # See /usr/share/doc/bash-doc/examples in the bash-doc package.  if [ -f ~/.bash_aliases ]; then     . ~/.bash_aliases fi   So I created a .bash_aliases file in my home directory, and added only this line:  alias apacheres ='/etc/init.d/apache2 restart'  When I started my terminal, I get this error message:  bash: alias: apacheres: not found bash: alias: =/etc/init.d/apache2 restart: not found   The shell has obviously read the file, otherwise it wouldn't know of its existence. I have checked the directory path and I can find the file apache2 there. What do I need to do to make this alias work? What’s the rule for adding “-er” vs. “-or” when forming an agent noun from a verb? What’s the rule to decide whether you add -er or whether you add -or when creating an agent noun from a verb?    Sometimes it’s -er:      read > reader hate > hater hit > hitter   But other times it’s -or:    meditate > meditator collect > collector Say I have two strings,  String s1 = "AbBaCca"; String s2 = "bac";   I want to perform a check returning that s2 is contained within s1. I can do this with:  return s1.contains(s2);   I am pretty sure that contains() is case sensitive, however I can't determine this for sure from reading the documentation. If it is then I suppose my best method would be something like:  return s1.toLowerCase().contains(s2.toLowerCase());   All this aside, is there another (possibly better) way to accomplish this without caring about case-sensitivity? How to prove that exponential grows faster than polynomial? The habitual past "would" versus "simple past" tense One can use the habitual past 'would' to describe past events. For example one could say,      Jim would spend weekends at the farm. He would cut weed and water the plants.   One can also use the simple past.       Jim spent weekends at the farm. He cut weed and watered the plants.   Would the use of habitual past be considered "weak" writing or prose, as opposed to the simple past?  How does the simple past in the following sentence "He would cut weed and water the plants." without any preceding time expression e.g. "spend weekends"  make a distinction between a repeated action/habit and a one-time action of cutting weeds and watering plants?