Integral of $\frac{x^2+1}{(1-x^2)\sqrt{1+x^4}}$
How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$
We know that if a finite field $F$ has characteristic $p$ (prime), then $F$ has cardinality $p^r$ where $r = [F:\mathbb{F}_p]$.   I'm now trying to say something about the possible cardinalities of subfields of $F$. I can see that there is a subfield of cardinality $p^s$ for each $s$ that divides $r$, given by the fixed field of the group generated by $\phi^s$, where $\phi$ is the Frobenius automorphism.  Now suppose $K$ is a subfield of $F$. Then (since both are additive groups), Lagrange gives us that $|K|$ divides $|F|$, so $|K| = p^t$ for some $1 \leq t \leq r $ (alternatively, $K$ contains $\mathbb{F}_p$ and so is a vector space over $\mathbb{F}_p$ and is thus isomorphic to $\mathbb{F}_p^t$, where $t = [K:\mathbb{F}_p]$). By considering the multiplicative group of units of $K$ and $F$ respectively, we get that $ p^t - 1$ divides $p^r -1$. I want to make the leap to $t|r$, but I'm failing to see why this needs to be true. Any help would be appreciated. How were the solar masses and distance of the GW150914 merger event calculated from the signal? How can I duplicate a mesh along a curve? I want to outline a curve in the form of text with a mesh (UV-sphere, cube etc.). How can I do that while saving mesh geometry without using the Curve modifier? I have what seems to me a very simple proof that $A_{n}$ is the only subgroup of $S_{n}$ of index 2.  Since I've seen other people prove it with what feel like really complicated methods (), I'm wondering if I've overlooked something.  Proof: Let $H$ be any subgroup of index 2 in $S_{n}$.  Then $H\cap A_{n}$ is a normal subgroup of $A_{n}$ and since $A_{n}$ is simple, $H$ is either the trivial or the improper subgroup of $A_{n}$.  If it's the trivial subgroup it doesn't have index 2 in $S_{n}$, and otherwise the theorem is proved. I need to somehow retrieve the client's IP address using JavaScript; no server side code, not even SSI.  However, I'm not against using a free 3rd party script/service. Export nodes from Drupal 4? How to favorite a question from Android app? What is the quickest way to send some text from desktop\laptop to iPhone? I usually send emails to myself, but I wonder if there is a better way to share something quickly from my PC\laptop to iPhone. How to Achieve this Requirement ???Can i use Jave Script , CSS and Visual force page? There is a sentence:     No sooner had he sat down than he fell asleep.   I just do not understand, is this an inversion? And if so, I still do not understand the sentence. Sci-fi novel involving living forever through teleporter copies Objective-C Static Class Level variables Badass and BAMF are both modern words with approximately the same meaning: "Someone who is awesome to an extreme level, thereby leveraging unquestionable authority." Is there another noun or title that would convey the same level of awe, admiration and toughness that calling someone a badass does, without the vulgarity?  Thanks for any help. :) Why are moderators now required to report underage users? Just today (December 2, 2020), a  was enacted, which requires moderators to report users that they find to be underage:  If you find a user claiming to be under 16 escalate it to us through the &quot;Contact Community Team&quot; button and select the &quot;underage user&quot; template - this will flag the on-duty CM for action so we can investigate and, if deemed necessary, delete that account and all personal data associated with it. [...] This policy only applies to sites where you are a moderator. If you see things on a site where you’re not, there is no obligation to escalate.  Being part of the , moderators are required to follow the instructions there, and not following them is considered a violation of the moderator agreement. However, this contradicts a , which states (bold added):  Users and moderators are not legally responsible for underage users and are under no obligation to report them to us. If out of concern you would like to report it, [...]  As part of the re-enaction of the policy on December 2, the text &quot;and moderators&quot; in the above quote was edited out of that statement. Why are moderators now required as per policy to report underage users that they come across? Is there a legal explanation as to why SE's stance on this changed? I'm aware of  from a then-employee, which stated that moderators are expected, but not required as a condition of remaining a moderator, to report underage users. However, the new policy requires reporting them. Why? An example of a case where a moderator may not want to report an underage user is if the user claims to be from an EU country that has reduced the required age from 16. While SE's Terms of Use specify a blanket limit of 16 to comply with regulations in all EU countries, individual member states do have the ability to reduce the limit, so long as it's not below 13. There is no law violation if, say, a 15-year-old user from an EU country that has reduced the age limit to 14 registers, so a moderator may not feel like reporting them (since in that case they're merely enforcing the Terms of Use, which they're not otherwise expected to do).  The policy was recanted a few days after it was originally enacted on October 15, 2020, but was enacted again on December 2 after minor revisions following a review among moderators. The same question still applies to the revised policy, however. Is it OK for OSB3 boards to get wet during construction? Dilemma: Present Perfect Or Perfect Continuous? Pivot rows into multiple columns Fact  The intersection of a closed set $A$ and a compact set $B$ is closed.  Background  I wanted to prove that the intersection of closed and compact sets is compact. There might be other ways, but I am not interested in them for now and am looking to prove the above statement.  Relevant Facts  I don't have much idea what to use. $A^c$ is open. Every cover $\cup_{i\ge1}U_i$ of $B$ contains a subcover $\cup_{i=1}^kU_i$. I might want to prove $A\cap B^c$ is open. Limit points may be another points to consider but I don't get any idea in that direction.  I also don't have that $B$ is closed otherwise it would be trivial.  Some Clarification  I am aware that compact sets are not closed in arbitrary topological spaces. The reason why I was motivated to believe tha intersection of closed and compact set is compact is true is: See Matt E.'s Answer      So if $K$ is compact in an arbitrary topological space (which is just to say that   it is a compact topological space when given its induced topology) and     $L$ is closed then $K \cap L$ is a closed subset of $K$ in its induced topology   Is it true if $A$ and $B$ are not whole spaces?  I would appreciate any hints. What makes Stealing Signs wrong? In the film Thor, Odin casts a spell on Mjolnir and strips Thor of his power.  He then banishes Thor to Midgard (Earth).  On Earth, Thor is seemingly just a strong and durable human.  Later in the film, he becomes worthy, and regains control of Mjolnir.  This then grants him his full power set and control over Mjolnir.  My question is, prior to this, could Thor control lightning all on his own?  Or is this purely an ability of Mjolnir?