Let $\mathbb F_q$ be the finite field of order $q$, where $q\equiv 1\pmod 4$ is some prime power. Let $\chi_4\colon\mathbb F_q^\times\to\mathbb C^\times$ be a multiplicative character of exact order 4 (so $\chi_4(\theta)=i$ for some generator $\theta$ of the cyclic multiplicative group $\mathbb F_q^\times$). In Neal Koblitz's "Introduction to Elliptic Curves and Modular Forms," Exercise 3 in Section II.2 says to show that $\chi_4(-4)=1$. To do this, it suggests showing that $\chi_4(4)=\chi_4(-1)=1$ if $q\equiv 1\pmod 8$ and $\chi_4(4)=\chi_4(-1)=-1$ if $q\equiv 5\pmod 8$.   I first tried proving that $\chi_4(4)=1$ if $q\equiv 1\pmod 8$. It would be nice if I could prove that $2=u^2$ for some $u\in\mathbb F_q^\times$ so that I could say $\chi_4(4)=\chi_4(u)^4=1$, but I got stuck with this. It dawned on me that I really don't know how to do computations in finite fields. How can I prove the given problem. More generally, is there any nice technique for handling computations like this?
Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I have no background in Algebraic Number Theory, but if someone explains in that language, I would like to see it and learn it as much as I can at this moment. References will also be appreciated!  Let $q$ be a power of prime $p$ such that $q \equiv 1 \mod 4$ and $p \nmid 2n$.  Question. Let $\chi_{4} : \mathbb{F}_{q}^{*} \rightarrow S^{1}$ be the character of order $4$, show that $\chi_{4}(-4) = 1$.  The hint in the book says we may show that $\chi_{4}(4) = \chi_{4}(-1) = 1$ for $q \equiv 1 \mod 8$ and $\chi_{4}(4) = \chi_{4}(-1) = -1$ for $q \equiv 5 \mod 8$.  If $q \equiv 1 \mod 8$, then $\chi_{4}(4) = \chi_{4}(2)^{2} = \chi_{2}(2) = \pm 1$, where $\chi_{j}$ denotes the character of order $j$. So I think $2$ must be quadratic residue in this case, but I cannot figure out why.
How to use an existing database with an Android application ng-options with simple array init Page Object Model or External property file for Selenium Object Repository I am a beginner in Selenium WebDriver (Java with TestNg). I'm trying to create a framework of my own.  My question here is what would be the better option to store my objects (basically all the webelements of my website under test), Page Object Model or External Property file? Why? Is there a better method than those? Please help. $m$ is the Lebesgue measure. I was thinking that: $\Bbb Q$ is dense and $m(\Bbb Q)=0&lt;1$ but it fails to be open, but maybe I could construct an open set from this fact, or also using that $(0,1)$ is a dense and open subset but $m(0,1)=1$.   Any hints or ideas will be very appreciated. Thanks. Why must a planet be in a star's "habitable zone" to have life? Why is notepad++ writing to my new document? I updated notepad++ and it started writing to the screen.  I quickly closed it not knowing what it was, but now I'm curious do I have a 'hacked' copy of notepad++? TAR 6 GB worth of files into separate 1.9 GB TAR files Are the square roots of all non-perfect squares irrational? $\left|x_{n+1} - x_n\right| < C\left|x_n - x_{n-1}\right|$ prove this is Cauchy Are there any galois fields which consist of product of two primes, as in $\mathrm{GF}(2\cdot 3) = \mathrm{GF}(6)$? Integral $\int\csc^3{x} \ dx$ I found these step which explain how to integrate $\csc^3{x} \ dx$. I understand everything, except the step I highlighted below.  How did we go from: $$\int\frac{\csc^2 x - \csc x \cot x}{\csc x - \cot x}\,dx%$$ to  $$\int \frac{d(-\cot x + \csc x)}{-\cot x + \csc x} \quad?$$  Thank you for your time! $$ \int \csc^3 x\,dx = \int\csc^2x \csc x\,dx$$ To integrate by parts, let $dv = \csc^2x$ and $u=\csc x$. Then $v=-\cot x$ and $du = -\cot x \csc x \,dx$. Integrating by parts, we have: $$\begin{align*} \int\csc^2 x \csc x \,dx &amp;= -\cot x \csc x - \int(-\cot x)(-\cot x\csc x\,dx)\\ &amp;= -\cot x \csc x - \int \cot^2 x \csc x\,dx\\ &amp;= -\cot x\csc x - \int(\csc^2x - 1)\csc x\,dx &amp;\text{(since }\cot^2 x = \csc^2-1\text{)}\\ &amp;= -\cot x \csc x - \int(\csc^3 x - \csc x)\,dx\\ &amp;= -\cot x\csc x - \int\csc^3 x\,dx + \int \csc x\,dx \end{align*}$$ From $$\int \csc^3 x\,dx = -\cot x\csc x - \int\csc^3 x\,dx + \int \csc x\,dx$$ we obtain $$\begin{align*} \int\csc^3x\,dx + \int\csc^3 x\,dx &amp;= -\cot x \csc x + \int\csc x\,dx\\ 2\int\csc^3 x\,dx &amp;= -\cot x\csc x + \int\csc x\,dx\\ \int\csc^3x\,dx &amp;= -\frac{1}{2}\cot x\csc x + \frac{1}{2}\int\csc x\,dx\\ &amp;=-\frac{1}{2}\cot x\csc x + \frac{1}{2}\int\frac{\csc x(\csc x - \cot x)}{\csc x - \cot x}\,dx\\ &amp;= -\frac{1}{2}\cot x \csc x + \frac{1}{2}\int\frac{\csc^2 x - \csc x\cot x}{\csc x - \cot x}\,dx\\ &amp;= -\frac{1}{2}\cot x \csc x + \frac{1}{2}\int\frac{d(-\cot x+\csc x)}{-\cot x +\csc x}\\ &amp;= -\frac{1}{2}\cot x\csc x + \frac{1}{2}\ln|\csc x - \cot x|+ C \end{align*}$$ Is it touch(1) from coreutils-8.4-37.el6.x86_64 or my brain which is broken?  $ touch abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:06.189751847 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:06.189751847 +0800 $ touch -a abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:17.374235446 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:17.374235446 +0800 $ touch --help | grep 'access time'   -a                     change only the access time   As you see, it was not only atime but also ctime which was updated by touch -a !?!?  The filesystem is ext4 over LVM if that makes any differences. When it comes to getting a tenure-track position where one has both a lot of resources and academic freedom to do what one wants to do (so having a tenure-track position in a top university might be more useful because one may get more resources at a top university).   Or in short, when one wants to maximize one's chances of becoming a "top scientist".  There are obviously numerous factors, and "rank/prestige" should not be considered as a factor in itself. That said, the period after one gets a PhD is an extremely important time for building up connections, and those are easier to get at a top university. "Hot" tab in is badly skewed and misleading - time to replace it with actual hot questions? Each site got a &quot;hot&quot; tab, where its contents are calculated based on . However take a look on the screenshots below and see for yourself that &quot;hot&quot; got nothing to do with many of the questions listed there. :    :    As you can see, there are questions with negative score, no answers, and even closed. I won't be surprised to see spam in there as well. This can't be any worse. What I suggest, instead of fixing the formula or patching it, simply have the hot tab show the same questions as the Hot Network Questions sidebar. (from the specific site only, of course.) Since the hot questions &quot;pool&quot; (as seen ) is limited to 100 for the whole network, either expand it or have the per-site list show questions with less hotness points - it would still be better than what we have now. How to show that $f'(x)<2f(x)$ This is question B4 in 1999 Putnam Exam. I would appreciate if somebody could help me with this.      Let $f(x),f'(x),f''(x),f'''(x)&gt;0$ , $f'''(x)$ is a continuous function and $f'''(x)&lt;f(x)$ on $\mathbb{R}$. Then show that   $$f'(x)&lt;2f(x),~ \forall x\in \mathbb{R}.$$ How can I set my system to suspend when inactive to a longer amount of time I run Mythtv on some Ubuntu 11.10 based machines.  In the power settings, I can choose from a list of time periods from 5 minutes up to 1 hour for the the machine to suspend when inactive.  I would like to set the suspend to occur at a longer interval, like 2 hours or 4.  Previously with Gnome, I would think that I could go into the gconf editor and find the key there, I am not sure with Unity where to override a setting like this.  Any suggestions? What is this style of text-as-image called? I'm seeing a trend (especially on pinterest) of these posters where the image and subject is the text. THe text itself isn't really that important either, it's mainly the typefaces used and composition.  Does this style have a name to it? Is it just called text-as-image? Books for starting with analysis I am interested in self-studying real analysis, and I was wondering which textbook I should pick up. I know all high school mathematics, I have read  (I did most of the exercises).  I have completed a computational calculus course which covered everything up to and including integration by parts (including the substitution method and Riemann sums) I am currently considering:    From what I have heard this is not very well suited for self-study and that while the exercises are extremely difficult, if you take the time they are worth the effort.    I have heard that while Spivak explains proofs in much more detail than Principles, it doesn't cover all of the material in the latter.    I don't know much about this. I have only seen some comments saying that it is an excellent introduction to analysis.    Extra clarification edit: I would prefer a book not to ''dumb down'' the material, something that would not hold my hand through every step, something that would force me to fill in the gaps myself instead of explaining every single step. That is why I am currently leaning towards Rudin, but before I decide I would still like some information on the book by Apostol. Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$.    I'm not sure how to start this proof, if it should be done algebraically or if I should try to use a proof by induction. Include sorting of accepted answer on profile I have few things on my mind:   I viewed my profile and I wanted to sort my answers by  "Accepted Answer, Votes" Wanted to know how many answers I've made that is marked as  the accepted answer. Sort by Most liked Question: I'm referring to my question who  have the most numbers of favorites. I wanted to sort the questions I created order by accepted or  unaccepted questions.    1 &amp; 2 seems useful to me but 3 &amp; 4 I don't think are that useful. But who knows, someone in this site might find it useful one day!  My POV: My answer is only sorted by Votes, I also want to see a list of accepted  answers on my profile probably not just to know how many accepted answers  I have but also to know what are the accepted answers that I have.