This is the last of the stream of number theory problems I have been looking at that I would like to discuss.     Let $p$ be an odd prime number and let $m$ be a positive integer.   Prove that any primitive root modulo $p^m$ is also a primitive root   modulo $p$.   My idea is as follows: Suppose $r$ is a primitive root modulo $p^m$. Then $r^{p^{m-1}(p-1)} \equiv 1 \mod{p^m}$, and therefore $$\left( r^{p-1} \right)^{p^{m-1}} \equiv 1 \mod p.$$ Can we conclude from this that $r^{p-1}\equiv \pm 1\mod p$, and since $p^{m-1}$ is odd, that $r^{p-1}\equiv 1 \mod p$?  How do I finish?  EDIT: Already answered at  . The chosen solution is very simple: If $r$ is a primitive root modulo $p^m$, then there exists an integer $k$ such that $r^k \equiv a \mod{p^m}$ for any $a$ relatively prime to $p^m$. Therefore $r^k \equiv a \mod p$. This can only be true if $r$ is a primitive root modulo $p$.
For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of  $p$   So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to show that.Can anybody help me this method?Any other type of prove is also welcomed.
What is the "canary thread" and why is it starving? The following occasionally appears in /var/log/syslog:  rtkit-daemon[1145]: The canary thread is apparently starving. Taking action. rtkit-daemon[1145]: Demoting known real-time threads. rtkit-daemon[1145]: Successfully demoted thread 1431 of process 1368 (n/a). rtkit-daemon[1145]: Successfully demoted thread 1430 of process 1368 (n/a). rtkit-daemon[1145]: Successfully demoted thread 1368 of process 1368 (n/a). rtkit-daemon[1145]: Demoted 3 threads.   What's going on here? Search with Auto-Complete (like Gmail / Intellisense) Can $\pi$ or $e$ be a root of a polynomial with algebraic coefficients? Since $\pi$ and $e$ are transcendental, neither can be the root of a polynomial with rational coefficients. However, it is easy to construct a polynomial transcendental coefficients (with $\pi$ or $e$ as one of it's roots), namely $(x-\pi)$ and $(x-e)$. However, is it possible to construct a finite polynomial (one of whose roots is $\pi$ or $e$) with only algebraic coefficients? (i.e. something such as $y=x^5-(\sqrt 2 -1)x^2+x$ .) A polynomial with only radical coefficients would be the best.  P.S. The polynomial doesn't necessarily need to have BOTH $\pi$ and $e$ as it's roots, just one is interesting enough. I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, there must exist a linear functional from $\ell_2$ into $\mathbb{R}$. However, I'm having trouble actually coming up with it.   I believe I'm supposed to find an unbounded function (although I'm not sure why an unbounded function is necessarily not continuous; some light in that regard would be appreciated too), so I thought of using the vectors $e^i$, which have all entries equal to zero, except for the $i$-th one. Then, you can define $f(e^i)=i$.  That'd be unbounded, but I'm not sure if it'd be linear, and even if it is, I'm not sure how to define it for all the other vectors in $\ell_2$.   A friend mentioned that at some point the question of whether the set $E=\{e^i:i\in\mathbb{Z}^+\}$ is a basis would come up, but I'm not sure what a basis has to do with continuity of $f$.   I'm just learning this topic for the first time, so bear with me please.   The space of sequences that are eventually zero (suggested by a few people) turned out to be exactly what I needed.  It also helped to cement the notions of Hamel basis, not continuous, etc. Why are local variables not initialized in Java? Change \headheight for a \pagestyle I have a document class that auto generates header from some user input. The problem is the first page should have a different page header than the others and the headline can be multi line. So \headheight is too small. Here is a demonstration of the problem:  \documentclass[twoside]{scrartcl} \newcommand{\longfirstpagehead}{% \begin{minipage}{10cm} some\\long\\text\\text\\text\\text\\text\\text \end{minipage}} \newcommand{\normalhead}{normal} \usepackage{scrpage2}  \newpagestyle{firstpage}{%     {\longfirstpagehead\hfill} %twoside links     {\longfirstpagehead\hfill} %twoside rechts     {\longfirstpagehead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %  \newpagestyle{normal}{%     {\normalhead\hfill} %twoside links     {\normalhead\hfill} %twoside rechts     {\normalhead\hfill} %oneside     (\textwidth,1pt) }{%     {\pagemark\hfill} %twoside links     {\hfill\pagemark} %twoside rechts     {\hfill\pagemark} %oneside } %    \pagestyle{normal} \AtBeginDocument{     \thispagestyle{firstpage}}  \usepackage{blindtext} \begin{document}     %\vspace*{1cm}     \blindtext      \blindtext     \pagebreak \end{document}   I tried to read the height with \settototalheight and:   add a \vspace at beginning of document (unnice hack):  still the bad box warning (of cause) first section has different alignment  manually change \headheight in pagestyle  text height and page foot is wrong (can't change this because some scoping?)  use Komascript in pagestyle  \recalctypearea warning  geometry in pagestyle  pretty much the same problem like the manual attemp    So I would like to change the \headheight for the pagestyle firstpage, but the page should still have the same page margins, etc. The problem is that this has to work with \thispagestyle. (Better Ideas are also welcome) Something like \thispagegeometry in geometry would be great. ;) Headphones don't work after update I updated to Ubuntu 15.04 from Ubuntu 14.10 a few days ago, and now my Logitech G230 headphones don't receive any sound. How can I fix this? PulseAudio Volume Control detects my headphones, and says they're working, but no sound comes through.  EDIT: Using alsamixer worked at first, but now it doesn't, and I get frequent bursts of static through my headphones now. Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$ Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$?  That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$? Intersection of open affines can be covered by open sets distinguished in *both*affines Suppose $X$ is an arbitrary scheme and $U \cong \operatorname{Spec} A$ and $V \cong \operatorname{Spec} B$ are affine upon subsets of $X$.  It's not true in general that $U \cap V$ is affine, so if we want to prove basic results about general schemes (e.g. the stuff in section II.3 of Hartshorne), we need some technical means of understanding what goes on on the intersections of open affines.  I've heard that this can be done by covering $U \cap V$ with open sets which are distinguished in both $U$ and $V$, i.e. sets $W \subseteq U \cap V$ with $W \cong \operatorname{Spec} A_f \cong \operatorname{Spec} B_g$ for some $f \in A$ and $g \in B$.  I'm having trouble actually proving this.    So, questions:   Is this actually true? If so, how are these sets constructed? Traverse a list in reverse order in Python So I can start from len(collection) and end in collection[0].  I also want to be able to access the loop index. How to connect Android with MySQL using Mysql JDBC driver Why are low-rep users rate-limited when posting questions? I just encountered the following error:  Oops! Your question couldn't be submitted because: users with less than 125 reputation can only post questions every 20 minutes; try again later.  Why do you have this rule? Why 20 minutes? Where can I find stock or custom ROMs for my Android device? For QA workstations preparation (Windows-based PCs) our IT department uses typical batch deployment tools. I'm not at all interested in the details of such process, but IT guys ask me:  What typical set of software you need for everyday manual testing tasks? We could include these in our deployment so that you guys could save more time for testing, not for environment prep &amp; tuning.  Sure thing we do test various range of projects, from web applications to standalone solutions, and each application requires its own set of tools for manual testing, e.g.:  Browsers Virtualization utilities DB clients Screens capturing tools FTP clients XML editors etc.  However, and that's for sure - there's a list of tools every QA MUST have installed on its PC, regardless of project types under testing. My area of experience is mostly standalone / complex web apps, that's why my vision on the task is a bit subjective. So, your suggestions of what are tools / software an average QA specialist can't live without will be highly appreciated. 2 assumptions:  Manual testing is a point of interest. Let's limit the list to not more than 15 (better 10) items.  Thanks in advance! Occasionally I see constructs like this in English sentences:        Not only is this true in this case, but ...      What puzzles me is the "is this" part of it. The sentence is not a question, yet it seems to have a question-like order. Isn't this correct/preferred:        Not only this is true in this case, but ...      Are there other cases where non-question sentences contain question-like parts (so to speak)? Is it possible to use a bluetooth headset mic while having High quality sound I have a Plantronics BackBeat Sense SE bluetooth headset. When Connecting it to Windows 10, it installs two profiles in Playback devices list:    PLT_BBTSENSE Hands-Free. PLT_BBTSENSE Stereo.   The Second one (Stereo) is set to be the "Default Device" and the "Default Communication Device" on the system.   At this point the sound works great and normally. The problem happens when I want to use the mic. When I start any program that uses the mic (recorder, chat, VoIP Calls, gaming, etc.) The sound suddenly stops working And I can only use the mic until I stop the recorder or the call.   The thing that kind of fixes the problem is setting the (Hands-Free) as the default device. I can hear the sound when I do that and it works with the mic but it becomes Very bad quality.  What I became to understand is that bluetooth had different profiles for sound and mic: A2DP and HSP. The first handle the voice(High Quality) but the latter handles the mic and reduces the sound quality.  Is this really the case? Is there anyway to make them work simultaneously? Is there any solution to this problem at all? I'm running a witcheling, a changeling witch who, because of a GM's homebrew magic ring, can transform into a .  What effects does this pixie transformation have? Specifically...   Since I'm now Small, do I apply +1 to AC, +1 to attack rolls, –1 to CMB and CMD and +4 to Stealth? Am I able to cast spells while transformed? Do I lose my racial benefits while transformed? Am I still able to use weapons such as staff, longsword, or greatsword? Can the Force be used to fly? In The Empire Strikes Back Yoda uses the Force to lift Luke's X-Wing out of the water and put it on land.  In Attack of the Clones and in many instances in the second series of The Clone Wars, Anakin (and others) use the Force to jump higher than they could otherwise, or to jump from very high locations, but I don't think I've ever seen the Force used for outright self-levitation.  But if Yoda could lift the X-Wing, shouldn't a strong Jedi be able to lift himself and, essentially, fly with the help of the Force? Indentation shortcuts in Visual Studio If I run:  sudo apt-get --yes install postfix    or  sudo bash -c 'yes | apt-get --yes install postfix'   an interactive prompt appears to configure postfix. I need to automate the installation of postfix (I can modify the configuration files after the install).  Is there some magic that lets me install postfix (and other package) without human intervention?