Why is $K:=\{x+\sqrt{2}y : x,y\in \mathbb{Q}\}$ a field?
Show that the set $\mathbb{Q}[\sqrt{2}] = \{a + b \sqrt{2} \mid a, b \in \mathbb{Q}\}$ is a field with the usual multiplication and addition.
Can long term sleeping at day damage your health? I've passed half or more of my life, sleeping in very late and often after the sun rises. Now I believe that I'm already feeling its effect. Some of my hormones are under the ideal levels and I have a deficiency in vitamin D.  What are the effects of sleeping during the day and not at night for work reasons or just habit? Is there any effect that can't be reversed simply by sleeping at night? (Assuming that you are sleeping enough time per day like 7-8 hours)  Could this sleep pattern be related to endocrine ills and other ills? Can you nest html forms? Is it possible to nest html forms like this  &lt;form name="mainForm"&gt;   &lt;form name="subForm"&gt;   &lt;/form&gt; &lt;/form&gt;   so that both forms work? My friend is having problems with this, a part of the subForm works, while another part of it does not. Fast calculation for $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$ I want to show that $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$, but is there a faster method than finding the contour and doing all computations?  Otherwise my idea is to do the substitution $x=e^t$, integral than changes to $\int _{-\infty}^{\infty}\frac{t e^t}{1+e^{2t}}dt$. Next step is to take the contour $-r,r,r+i\pi,-r+i\pi$ and integrate over it... I am not very familiar with databases and the theories behind how they work.  Is it any slower from a performance standpoint (inserting/updating/querying) to use Strings for Primary Keys than integers? Simple Power Series Expansion for Problems similar to $f = (1 + \epsilon \,x)^{1/\epsilon}$ I was flicking through a book on perturbation methods and saw a simple question asking the reader to expand the following expression for $f$ in a power series (up to the first 2 terms):  $f = (1 + \epsilon \,x)^{1/\epsilon}$, where $\epsilon$ is a small parameter. I'm sure this is very simple, but I wasn't certain about the best way to approach this. A quick look at mathematica tells me the solution is $e^x - \frac{1}{2} (e^x x^2) \,\epsilon + ...$. How would I go about getting this answer - and more importantly, how would I systematically find series expansions for problems similar to this one? Consider the sentence      E-books are on the rise, but they haven't suppressed paper books though.    This usage seems to be quite common, but when I learned English  I was taught to use "however" where I now read "though". My questions on this use of "though" are:     Is this a new trend? Is it restricted to American English? Is "though"  considered more colloquial than "however" ?    And on a syntactic level: Do you put a comma before "though"? What should I do when a hacker gives me hacked money? I was peacefully playing GTA Online (yeah, was killing someone coming out of ls customs) then a hacker came and killed me using explosive ammos, I called him a hacker and he teleported to me and gave me 790k. I know that people get banned for receiving this kind of money, what should I do to avoid the ban? Can someone please give me the prerequisites and mathematics required for string theory? Are there some good references to study it, both online and in a book? Please consider I am a newbie in string theory, and only about up to the level explained in Hawking's Brief History of Time and Penrose's Road to Reality. I would prefer something which does not omit rigor, but builds it up.Also, are there good online lectures on this?  (I have covered most of GR, but I am not really comfortable with advanced Quantum mechanics. I have studied QM only from Griffith's book, and a few McGraw hill lecture notes.) I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question.  Prove directly from the definition of Ito's integral that   $$\int_0^t B_s^2 dB_s=\frac{1}{3}B_t^3-\int_0^t B_s ds$$  I have tried for a couple of hours but I simply cannot prove it . I would be grateful if somebody could help me out , even a hint would be very helpful Site.createPortalUser() returning null in test context bilinear form - proof I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$  is a bilinear form, that is, inter alia, the condition: $f(x+y,z)=f(x,z)+f(y,z)$  I have: $f(x+y,z)=f\left({\displaystyle \sum_{i=1}^{n}}x_{i}e_{i}+{\displaystyle \sum_{k=1}^{n}}y_{k}e_{k},{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=f\left({\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{k=1}^{n}}\left(x_{i}e_{i}+y_{k}e_{k}\right),{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=\ldots$  and what's next?  Help! How can I determine what is taking up so much space? Can you cast a 9th level spell if your multiclass spell slots have 9th level spell slots? Big-O proof showing that t(n) is O(1) Let t(n) = $\sqrt{n^2+100n} - n$  Show that $t(n)$ is $O(1).$  I tried solving this using the formal definition of O( ) but am not getting anywhere. Quite confused on how to tackle this question. A white number inside a black circle. How to make a white number inside a black circle (the picture below)? I want to change numbers inside the black circle, of course.  I use Share LaTeX. I have read some articles here but anyone are not what I would like to get. I would like to get this  (the picture below) - the sixth answer.   Thanks for help. Finding minimum-area-rectangle for given points? How can I enable Hardmode? What items do I need to make my world Hard Mode? Can a Linux user belong to more than one group?  If it is allowed, then one use may access files from two groups, that would be quite nice!  If not, is there any alternative way to have this function?  This issue is not the same as  or Intervals are connected and the only connected sets in $\mathbb{R}$ As the topic, prove that Intervals are connected and only connected  in $\mathbb{R}$. I know what is the definition of connected set. But not sure how to prove that. How to disable notifications for a single mailbox in Mail? When managing multiple mailboxes, is it possible to disable the notifications for certain mailboxes?