Let $f(X)=X^6+X^3+1\in\mathbb Q[X]$. I need to find a splitting field $L$ for this polynomial and the degree of $[L:\mathbb Q]$.  $f$ is irreducible with $f(X+1)$ and $p=3$ to apply Eisenstein. I know that then $\mathbb Q[X]/(f(X))$ is a field in which $f$ has a root (how can I find out what the root is?). I'm supposed to inductively find more splitting fields and adjoin them in the end to get $L$, but how is that done concretely?
I am trying to find the splitting field of $x^6+x^3+1$ over $\mathbb{Q}$.  Finding the roots of the polynomial is easy (substituting $x^3=t$ , finding the two roots of the polynomial in $t$ and then taking a 3-rd root from each one). The roots can be seen  [if there is a more elegant way of finding the roots it will be nice to hear]  Is is true the that the splitting field is $\mathbb{Q}((-1)^\frac{1}{9})$ ? I think so from the way the roots look, but I am unsure.  Also, I am having trouble finding the minimal polynomial of $(-1)^\frac{1}{9}$, it seems that it would be a polynomial of degree 9, but of course the degree can't be more than 6...can someone please help with this ?
The layman's question. Is there a maximum temperature that can be reached by air on Earth due to global warming? For example, one that would prevent living organisms from functioning? I read somewhere that with the increase of the sun's brightness by 10%, the average temperature on our planet will be 47 C. This, however, only for about 600 million years. Would global warming enable reaching such a high average temperature? What is the right way to indent your source code? What processing is done on RAW files in the camera? Recently I came across a debate about whether it's fair to call RAW 'unprocessed' or 'unaltered' sensor data of a DSLR. As far as I know, the analog sensor data is processed into a digital RAW file using the ISO setting, a process during which the Bayer-sensor data is demosaiced as well. However, those are just the necessary steps for saving a digital file while maintaining maximum quality.  Are those assumptions correct? 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If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative. If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.  I need to proof this theorem and I have something like this below. However, I do not know how to continue this proof.   $(x+y)^{2}-(x+y)=x^{2}+xy+yx+y^{2}-x-y=(x^{2}-x)+(y^{2}-y)+xy+yx\in Z(R)$  so $xy+yx\in Z(R)$ Obi-Wan left Anakin for dead, and one movie later (in A New Hope) he was saying Anakin had been killed by Darth Vader.  What exactly did Obi-Wan know about Anakin and Darth Vader before A New Hope started? Should a toggle button show its current state or the state to which it will change? I have a quick question about buttons that toggle between two states. (Think Play/Pause, or Shuffle/Regular Play.) As the title says, should the toggle show it's current state or the state to which it will transition?  I think people are used to the Play/Pause convention. But the Shuffle/Regular play might be more confusing if you show the transition state instead of current. For example, the built-in music player on the Xbox 360 does it this way: when it's in shuffle mode, it shows the icon for direct play and vice versa and it always confuses me (am I in shuffle mode or straight play).  I see it this way: Play/Pause is more like action as in begin playing or pause playing. Yes behind the scenes it is a state transition but to the user, there is an action. Whereas Shuffle/Straight Play is an option and it's best to show the current state (and possibly have only one icon and change button to show that the option is enabled/disabled). Thoughts? How do I close a connection early? I'm attempting to do an AJAX call (via JQuery) that will initiate a fairly long process. I'd like the script to simply send a response indicating that the process has started, but JQuery won't return the response until the PHP script is done running.  I've tried this with a "close" header (below), and also with output buffering; neither seems to work. Any guesses? or is this something I need to do in JQuery?  &lt;?php  echo( "We'll email you as soon as this is done." );  header( "Connection: Close" );  // do some stuff that will take a while  mail( 'dude@thatplace.com', "okay I'm done", 'Yup, all done.' );  ?&gt; Standard library sort and user defined types Because of past conflicts, I don't believe my advisor will write me a good LOR for postdoc positions. What should I do? cgi and tomcat im trying to run a cgi script (.cgi) with tomcat. I am getting the below error and cant find out whats wrong. I know i should really use apache and mod proxy but this really isnt my area of expertise so im taking the easy way out!  Thanks for any help.   java.io.IOException: Cannot run program "perl" (in directory "C:\Java\tomcat\webapps\my_app_name\WEB-INF\cgi"): CreateProcess error=2, The system cannot find the file specified     java.lang.ProcessBuilder.start(ProcessBuilder.java:459)     java.lang.Runtime.exec(Runtime.java:593)     java.lang.Runtime.exec(Runtime.java:431)     org.apache.catalina.servlets.CGIServlet$CGIRunner.run(CGIServlet.java:1705)     org.apache.catalina.servlets.CGIServlet.doGet(CGIServlet.java:597)     javax.servlet.http.HttpServlet.service(HttpServlet.java:627)     javax.servlet.http.HttpServlet.service(HttpServlet.java:729)     org.tuckey.web.filters.urlrewrite.UrlRewriteFilter.doFilter(UrlRewriteFilter.java:738)     org.apache.struts2.dispatcher.FilterDispatcher.doFilter(FilterDispatcher.java:416) Op Amps: how do they know whether the supply is split? Most op amps can operate in single and split supply configurations. However, I cannot see how this is possible.  If I connect voltages \$V_\text{SY}^+\$ and \$V_\text{SY}^-\$ to the positive and negative supply pins of a op amp, and \$V_+\$ and \$V_-\$ to the non-inverting and inverting inputs, then the output is (if \$A_\text{OL}\$ is the open loop gain of the op amp): $$V_\text{out}=A_\text{OL}(V_+-V_-)+V_\text{SY}^-\tag{Single-supply}~\text{ }~~\text{ }~~\text{ }~~\text{ }~~\text{ }~~\text{ }$$ The above equation is from personal experience using op amps in single supply (i.e. \$V_\text{SY}^- = \text{GND}, V_\text{SY}^+ &gt; \text{GND}\$).  In split supply, we can use the following diagram from MIT's 6.002 introduction to electronics lecture video:  $$V_\text{out}=A_\text{OL}(V_+-V_-)+(V_\text{SY}^+-V_\text{SY}^-)/2\tag{Split-supply}$$  These two equations, however, are mutually inconsistent: If I take an op amp and hook up a single voltage source between its supply pins, how does the op amp know that the resulting voltage is not split between two sources?   It must know - but how does it know? Why is this true?  $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$ Differentiability implying continuity Theorem: If $f$ is finitely differentiable at $c$, then $f$ is also continuous at $c$.  But according to the definition, for a function to be differentiable at $c$, it need not even be defined at $c$, just that it should have a finite value in the vicinity of $c$.    My question: If function is not defined at $c$, how can it be continuous at $c$?  For example, say we have a straight horizontal line function which is broken at $x=1$, the function is defined at all the points except $x=1$, it has a finite value at all the points except $x=1$, since it is having a value in the nearby region of $x=1$ we say that it is differentiable at $x=1$ but can we say that it is continuous at $x=1$? I want to talk to someone about the house that my wife and I own.  Saying, for example, "My wife's and my house is awesome," sounds a bit funny to me.  What's the best way to express this?  Clarification  I'm asking specifically about the grammar of multiple nouns in possessive form.  I'm particularly curious if it's possible to do this with a first-person pronoun (me).  I am capable of rephrasing this in other ways - my question is not how to express the idea, but about this particular grammatical construction, if it is even legal. Now, that Ubuntu 17.10 is out, and Unity 7 is replaced with Gnome 3, is there a way to install and use Unity7?   I know Unity is still in the repositories, so it should be possible, right? How to insert text before the first line of a file?