Find Polynomials with Integer Coefficients with Particular Roots Is there any simple way to find a polynomial with integer coefficients so that ($x=\sqrt{2} +\sqrt{3}$) is one of its roots? I know one way is to get rid of all the square roots in the equation to be left with rational numbers, but is there any other simple way?
Minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$ I had an example in the book given as follows:        Find the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$ .       Solution:  $~~~~~$$(\sqrt{2}+\sqrt{3})^2=5+2 \sqrt6$      $(\sqrt{2}+\sqrt{3})^4=49+20 \sqrt6$       Then $(\sqrt{2}+\sqrt{3})^4-10(\sqrt{2}+\sqrt{3})^2+1=0.$       Thus $a=\sqrt{2}+\sqrt{3}$ satisfies $f(x)=x^4-10x^2+1$ over $\mathbb Q$.       Let $p(x)$ be the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$.Then $(\sqrt{2}-\sqrt{3}),(-\sqrt{2}+\sqrt{3}),(-\sqrt{2}-\sqrt{3})$ are also roots of $p(x).$   So degree of $p(x)$ is atleast $4$.      But $f(a)=0 $ and $f(x) \in \mathbb Q[x] \implies p(x)$ divides $f(x)$ .      So $f(x)$ is minimal polynomial of $\sqrt{2}+\sqrt{3}.$   But I can't get the step  how are $(\sqrt{2}-\sqrt{3}),(-\sqrt{2}+\sqrt{3}),(-\sqrt{2}-\sqrt{3})$ also roots of $p(x).$   Kindly help with this.
Female main character in an ice world earning gems/runes Here's what I remember:   The main character's female (99% sure) There is one part where she has to cross a big tundra and she has to kill a big ice creature The kids in the story go to a school-like place where they train for something. I'm pretty sure they earn gems or runes or something like that There's one part where the main character has to play a piece of music perfectly to earn her gem/rune thing It's definitely a series; young-adult (I read them in middle school)   Thanks! If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$ Calculate the uncertainty in velocity of electron I got this question in one of my periodic test in coaching, I (seemingly) solved the question but later recognised what mess I did.  So, the question is :      Calculate the uncertainty in velocity of an electron. If the uncertainty in its position is ${10^ {-15}} \; \mathrm m$.   By simple calculations one can see that the answer would be ${5.79\times{10^{10}}} \; \mathrm{m/s}$. However, as one can see, the uncertainty in velocity is greater than the velocity of light then how is that possible or there is some other way to handle such problems.  P.S. I know the concept of dynamic mass but I am not very familiar with it. (I think that would work here, but don't know how?) HP laptop F1-12 keys inverted with Fn I recently bought an HP laptop. I'm pleased with it, save for one thing.  All the F1-12 keys are inverted. I mean the following:  All the F-keys have a symbol on it. Normally you would press Fn + your to use it. On this laptop, it's reversed. The symbol is what you get when you press the button normally and you have to press Fn + F-key to get the usual action.  So, if I want to refresh a webpage, instead of just F5, I have to do Fn + F5  I want to change this to regular behavior. Anybody know how I can do this?  Running Win7 Home premium, 64-bit. What should I do when my neural network doesn't learn? From reading the rules it would appear there are two kinds of damage and then straight loss of life:     118.3 If an effect causes a player to gain or lose life, that player's life total is adjusted accordingly.   From reading that I would guess that effects like Extort are not sources of damage. You simply lose the life.      119.2a Damage may be dealt as a result of combat. Each attacking and blocking creature deals combat damage equal to its power during the combat damage step.   So damage from creatures during combat is combat damage.     119.2b Damage may be dealt as an effect of a spell or ability. The spell or ability will specify which object deals that damage.   And that is direct damage from an object.  So the arguments are these:  Does loss of life as outlined by rule 118.3 (at top) count as being dealt damage? Do triggered abilities that redirect or reduce damage effect loss of life?  My second related question: You put  on a  and tap her to deal one damage to your opponent. Am I right in saying that because the damage was direct damage (not dealt during combat) then the triggered ability of preventing the combat phase of the damaged player is not triggered? The Tree of Life in Evangelion I've noticed references to the Kabbalah in a couple of anime I've watched.  One of them is Neon Genesis Evangelion.  Neon Genesis Evangelion has it in the sky in both the opening credits and in the series itself:     I also remember it was on the ceiling at NERV, but the best screencap I could find of that was too tiny to be visible.  So, my question is, Is there any deeper meaning to the use of the Kabbalistic tree of life, or is it just used because it looks cool? is there a way to find suitable ratio for matches to controls (using "ratio" in Matchit package in R or other programs) I am using matchit package in R to find a control set for my treated units and I am testing the model with different ratio to get suitable control set. Is there a way to find suitable ratio and I will use that to find my control set. Cleaning wax and pesticide from apple skin I tried a small experiment. I posted a post on Facebook at exactly: 21:01 and waited to see how many hours ago it will say. When it was 22:50 it still said that my post was posted 1 hour ago.  Though some days back on this forum some people told me that when something started at 21:01 and it is 22:50 that I should say it started 2 hours ago.  So whom should I trust now?  All social networks like Facebook, Instagram and YouTube etc. use the system where it would say 1 hour ago even if it was 119 minutes already, so meaning they will never say it was 2 hours ago unless at least 120 minutes passed.  This was the way I used to express 1 hour ago as well. However, since I heard the answers from many other people I feel unsure about it right now...  I am talking about casual speech with friends etc. In such scenario. What would you say? 1 or 2 hours ago? Is Facebook/YouTube/Instagram wrong or?   if we decide to agree on the theory that if it is more than 1 hour and 40 minutes it is a good idea to round to 2 hours ago.   So what time span would you imagine if something happened let say 10:10, now it is 11:50.  If somebody said to me that something happened 2 hours ago and I looked on time saying 11:50 I would image that it happened some time between. (9:30-9:50)  That is why it feels so weird for me to say 2 hours ago thought it has not been more two hours yet.   I dont know why but more logical for me it feels to say 1 hour ago though it can already be more than 100 minutes  and still just say that it happened an hour ago when talking with friends, teachers, whoever in casual speech unless it is something super important i would say exactly hours and minutes.  How do you say it in your life? How do you express time in such situations? Do you express time the same way as Facebook or?   I know that some of you may think this is some kind of joke, unfortunately it is not and I am going crazy and depressed without knowing system for this... It has been already 2 months and I still feel overwhelmed with this ideas and questions... I cannot help myself, please help me, please.  Thanks for your help. Can I travel to the UK unaccompanied on an accompanied child visa now that I am over 18? Let's say I have a generic member in a class or method, like so: public class Foo&lt;T&gt; {     public List&lt;T&gt; Bar { get; set; }          public void Baz()     {         // get type of T     }    }  When I instantiate the class, the T becomes MyTypeObject1, so the class has a generic list property: List&lt;MyTypeObject1&gt;. The same applies to a generic method in a non-generic class: public class Foo {     public void Bar&lt;T&gt;()     {         var baz = new List&lt;T&gt;();                  // get type of T     } }  I would like to know what type of objects the list of my class contains. So what type of T does the list property called Bar or the local variable baz contain? I cannot do Bar[0].GetType(), because the list might contain zero elements. How can I do it? Which objects can I use in a finalizer method? If you open any of localized meta site in browser's private mode, e.g.  you will see untranslated text on the top right corner:        This message is not found in Transifex, hence unable to localize.  Related bug report on ruSO.meta: I'm having serious problems installing the Broadcom drivers for Ubuntu. It worked perfectly on my previous version, but now, it is impossible.   What are the steps to install Broadcom wireless drivers for a BCM43xx card?  I'm a user with no advance knowledge in Linux, so I would need clear explanations on how to make, compile, etc.    lspci -vnn | grep Network showed:   Broadcom Corporation BCM4322 802.11a/b/g/n Wireless LAN Controller [14e4:432b]      iwconfig showed:  lo        no wireless extensions. eth0      no wireless extensions.   NOTE: Answer below is updated every time new information is added and confirmed working. How do I set an environment variable on the command line and have it appear in commands? If I run  export TEST=foo echo $TEST   It outputs foo.  If I run  TEST=foo echo $TEST   It does not. How can I get this functionality without using export or a script? Does editing still bump a question? Many times in the past I've edited a somewhat recent answer—say, from 30 minutes ago—and noticed that the concomitant question didn't bump to the top.  I just confirmed it with ; the question is currently sorted in the JavaScript page based on when it was asked, not my edit.  I've never made edits just to bump a question to whore rep—my original answer was wrong which was why I fixed it—but my understanding is that the CW self-edit trigger exists to prevent  edit-then-bump-then-score-rep—profit! activity.    If non-ancient edits no longer bump a question, should the self-edit CW trigger be brought inline with this?  Should an answer be made CW if you make 10 edits that actually bump the question?  and to avoid any implication of impropriety, I'm already capped on SO, so I'm not looking for any sort of "meta effect" on my answer Converting string into datetime Mouse Pointer Locator What are "first class" objects? When are objects or something else said to be "first class" in a given programming language, and why? In what do they differ from languages where they are not?  EDIT. When one says "everything is an object" (like in Python), does he indeed mean that "everything is first-class"?