This is related to my .  Suppose I have the polynomial $p(x) = x^2 - 2 \in \Bbb Q[x]$. This is irreducible over $\Bbb Q$ so we need to move up to the extension field $K/\Bbb Q = \Bbb Q(\sqrt{2})$. Now $p$ splits over $K$ into $(x - \sqrt{2})(x+\sqrt{2})$ and it can be shown that the Galois Group $\operatorname{Gal}(K/\Bbb Q) \cong \Bbb Z/2\Bbb Z$ since it contains only the identity automorphism and one involution that sends $\sqrt 2$ to its conjugate $-\sqrt 2$.   My question is this; what if I have a polynomial with ugly roots? Say, for instance,  $$p(x) = x^5 - x^3 - 2x^2 - 2x - 1$$  This was shown in my previous question to be irreducible over $\Bbb Q$ so the splitting field of $p$ is simply $\Bbb Q$ adjoin the roots of $p$. These roots are, however,   $$\alpha \approx 1.73469\ \ \ \beta \approx 0.79645e^{0.49411\pi i}\ \ \ \gamma \approx 0.9533e^{5.6494\pi i}$$  and the conjugates $\overline \beta, \overline \gamma$ (do we get the conjugates for free when we adjoin complex roots or do we explicitly have to adjoin them?).  Then the splitting field $F$ of $p$ is $\Bbb Q(\alpha, \beta, \gamma)$ and their conjugates (depending on the answer to the question just asked).  At this point I'm not sure where to go; I've been trying to list automorphisms that fix $\Bbb Q$, obviously the identity is included, it seems also that the autmorphisms  $$\varphi_1 : \begin{cases} \beta \to \overline \beta \\ \gamma \to \gamma \end{cases}\ \ \ \text{and}\ \ \ \varphi_2 : \begin{cases} \beta \to \beta\\ \gamma \to \overline \gamma \end{cases}$$  are in the galois group. Also, $\varphi_2 \circ \varphi_1 = \psi$ should be in the group and this sends both $\beta$ and $\gamma$ to their conjugates. Any composition of $\psi$ with either of the others I've listed gives the other. I'm stuck for finding more now though. Since conjugation is an involution automorphism these are self inverse.   I'm not sure where to go from here! Is what I've done thus far even correct or have I totally misunderstood the concepts I'm trying to understand here?
I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I  seem to be having some trouble with the technical stuff. A specific example would be how to find the Galois group of a given polynomial. I know some tricks, and I manage to solve some of those questions, but some not.  For example, one of the tests asks to find the Galois group of $x^{4}-4x+2$.  I can see that it is irreducible over $\mathbb Q$ (Eisenstein), but I have no clue as to how to find its Galois group over $\mathbb Q$. Can someone tell me how to do this? General techniques concerning this sort of problems are also welcome :).  Thanks!
I needed to edit a .config file (which I know for a fact is not creating these problems, just to clear that up) and it wouldn't let me save it, so I took ownership of /etc and all contents with the command chown -hR username /etc and that let me edit the .config file but now whenever I try to install any packages or use any sudo commands, it doesn't work  (I've checked for errors in the sudo file in sudoers.d and there was nothing wrong with that).   So how do I return it to the previous owner? How do I backup my Ubuntu Phone? Is it possible to identify what the parent program/app is from a running process? Graphing the amount of time a function takes If you cast Polymorph on a non beast creature then cast Awaken on their beast form how much carries over after polymorph ends?  Polymorph does not give any explicit information on what happens to your statistics when the polymorph ends:     The transformation lasts for the duration, or [drops to 0 hp]. [...] The target's game statistics, including mental ability scores, are replaced by the statistics of the chosen beast.  (Basic Rules, pg. 266)   So this leaves ambiguity when game statistics are altered during polymorph,   The charm effect of Awaken seems pretty clear from the  and reinforced by the specific wording referring to the target as a beast or plant during effects and a creature afterwards:     The awakened beast or plant is charmed by you [longer than polymorph.] When the charmed condition ends, the awakened creature chooses [...] (Basic Rules, pg. 216)   The other two effects of awakening (other than making a plant mobile) are less clear-cut:     The target gains an Intelligence of 10. The target also gains the ability to speak one language you know.  (Basic Rules, pg. 216)   I would guess the modification to intelligence would apply to the beast's stats and therefore end with polymorph.     as being independent of creature statistics and act more as the character's experience. I'd argue the experience of a stranger turning you into a mouse, then gifting you the ability - as a mouse - to speak a language you have never used before would be an experience you couldn't lose I'm not sure if the language and intelligence granted from awakening should remain linked.    These are two examples I can see this having significant impact on my campaign:   Meet an NPC I don't share a language with, Polymorph them then Awaken their beast form so they can speak my language. Can I now communicate with them when they turn back?  If their original intelligence was higher then 10 is it reduced to 10? Say I use find familiar to get an owl (classified as a fey) familiar, then polymorph it to an owl (classified as a beast) so my druid friend can cast Awaken on it, does my familiar keep an intelligence of 10 and/or the ability to speak?   So after reverting from an awoken polymorphed state would you keep the 10 intelligence and a new language, only the new language, or not keep either? Linux doesn't actually distinguish between processes and threads, and implements both as a data structure task_struct.  So what does Linux provide to some programs for them to tell threads of a process  from its child processes? For example,   Thanks. I am looking for a way to get the 15 character entity ID for a custom field on my Lead object in an Apex controller. I have tried this:  private String getEntityIdFor(String fieldName) {         Map&lt;String, Schema.SObjectField&gt; fieldMap = Schema.SObjectType.Lead.fields.getMap();         return fieldMap.get(fieldName).getDescribe().getId();     }    Of course, there is no getId() method, but this is the closest I got.   Thanks for any help. How to find a starting point for root finding in complex plane? The function $f(x) = x^2 +1$ has zeros in the complex plane at $x = +i$ and $x = -i$. Is there a real starting point for complex Newton's method such that the iterates converge to either of these zeros? Is there a complex starting point too?  I know that Newton's method for finding roots is $x_{n+1} = x_n - \frac {f(x)}{f'(x)}$. But I don't know how to find the proper starting point. To an an external observer it appears that time has stopped for photon. But this relation is reflexive, so for an observer travelling with the photon it appears the universe has stopped everywhere.  Is this right?  Space also gets distorted parallel to the direction of motion, but not perpendicular to it.   Does this mean that for an observer travelling with a photon sees spacetime as a flat plane?  Note 1: I'm using language vividly not literally when I say a photon experiences space and time. Not that I'm against idealist or panpyschist interpretations of matter or energy come to that.  Note 2: Since it's been pointed out that an inertial frame can't be boosted into the frame of a photon and moreover there is no limiting behaviour here - no matter what frame you are boosted into, light will always appear to travel at c. Nevertheless, when an external observer watches a train chasing a photon by gradually increasing its speed, he will note that the train speed is approaching the speed of the photon (but will never match it) and when he looks at the clock inside the train he will not too that it is gradually slowing down (but never actually stops). Using this picture, we can attempt the thought experiment - if not one that can be carried out - of what a world looks like to a photon. Why does Lang describe a field as a union and compositum of its subfields in this manner? I am reading Serge Lang's Algebra, revised 3rd edition, and on page 226, the author makes the following definitions and observations:   If $E,F \subset L$, the compositum of $E$ and $F$ in $L$ is defined to be the smallest subfield of $L$ containing both $E$ and $F$. It is denoted $EF$. If $k \subset E$ and $\alpha_1,\dots,\alpha_n \in E$, then $k(\alpha_1,\dots,\alpha_n)$ is defined to be the smallest subfield of $E$ containing $k$ and $\alpha_1,\dots,\alpha_n$. Observe that $$ E = \bigcup k(\alpha_1,\dots,\alpha_n), $$ where the union is taken over all finite families $\{ \alpha_1,\dots,\alpha_n \}$ of elements of $E$. The compositum of an arbitrary family of subfields of a field $L$ is defined as the smallest subfield of $L$ containing all fields in the family. $E$ is finitely generated over $k$ if there is a finite family $\{ \alpha_1,\dots,\alpha_n \}$ of elements of $E$ such that $E = k(\alpha_1,\dots,\alpha_n)$. Observe that $E$ is the compositum of all its finitely generated subfields over $k$.   My question is regarding the two observations (point nos. 3 and 6). It appears to me that it is enough to take union (and compositum) over all subfields generated by a single element. For example for point no. 3, I can write $$ E = \bigcup_{\alpha \in E} k(\alpha) $$ because $E$ is clearly contained in the RHS, and each $k(\alpha)$ is contained in $E$ and hence so is the union, implying that $E$ contains the RHS. Similarly for point no. 6.  So, why does Lang emphasise to take the union and compositum over all finitely generated subfields? Is there some perspective that he wishes to emphasise that I am missing? Any help in understanding this will be appreciated. How do I assign a value to a whole array of integers? Use SUM In Update Statement I am attempting to use SUM() in my SET statement for my UPDATE.  When I try this, I get an error of      Msg 157, Level 15, State 1, Line 234   An aggregate may not appear in the set list of an UPDATE statement.   What should I alter in my CTE so that I can run this update?    ;WITH CTE AS  (Select EC, CN, CID, EntryDate         FROM OPENQUERY(192.55.108.214,'select                                count(es.estimatenumber) As EC                               ,cm.an CN                               ,cm.am CID                               ,es.entrydate EntryDate                               from etalac es                                INNER JOIN bacalac cm                               ON es.CID = cm.am                               GROUP BY cm.an, cm.am, es.entrydate')) UPDATE d2016 SET TotalNumber = SUM(EC) FROM CTE cte INNER JOIN datafor16 d2016 ON cte.CID = ed2016.CID WHERE CAST(entrydate As Date) BETWEEN '01/01/2016' AND '01/22/2016'; If $f$ is continuous with $f(x) = f(2x),f(1) = 3$, then what is $ \int_{-1}^{1}f(f(x))\,dx$? If $f(x)$ is a continuous function such that $f(x) = f(2x)$ and $f(1) = 3\;,$ Then $\displaystyle \int_{-1}^{1}f(f(x))\,dx$   $\bf{My\; Try::}$ Here $-\infty &lt;x&lt;\infty$ and Given $f(x) = f(2x)$  So Using Recursively $$\displaystyle f(x) = f(2^1x)=f(2^2x)=..........=\lim_{n\rightarrow \infty}f\left(2^{n-1}x\right)$$   OR we can write it as $$f(x) = f\left(\frac{x}{2}\right)=f\left(\frac{x}{2^2}\right)=........=\lim_{n\rightarrow \infty}f\left(\frac{x}{2^{n-1}}\right)$$  Now How can i prove that $f(x)$ is a constant function.  Help me  Thanks Should I disable google maps scroll wheel zoom? In most applications of google maps the scroll wheel will zoom the map instead of scrolling it.  Personally I find this confusing, and after years of using google maps, both on the google maps web site and embedded google maps, I still often make the mistake of trying to scroll the map with scroll wheel and end up zooming it instead.  The google maps API allows me to disable scrollwheel zoom when I embed a map in my page.  The question is should I disable it?  On the one hand scroll wheel zoom is unconventional in the context of the web as a whole - in the vast majority of cases the scroll wheel scrolls things.  On the other hand scroll wheel zoom is established convention for maps.  It definitely would be a better experience for me if scroll wheel zooming was disabled, but what about the web browsing population in general?  I did some searching on whether I should disable scroll zoom and couldn't find any guidelines. So, to scroll zoom or not to scroll zoom? When changing fields, should I focus more on describing past research or connecting it to the new area in my statement of purpose? I want to change my field from molecular biophysics to theoretical ecology. While they are basically required some same basic background (physics, math, programming), the rest is quite irrelevant. In my current draft of this part, about a half is about this the problem and the result of the work, another half is for my contribution and what I've learnt. Having said that, should I focus less on describing the work and leave space for the thing that will be useful for the next research?  Below is copy from .     What have you already done? What problems have you solved, or at least worked on? What independent projects have you been part of? What were your key contributions? What did you learn? What did you teach the world? How do your results compare to what was already known? What original ideas are you most proud of? Be specific, technical, credible, and confident (but not arrogant). Refer the reader to your web page for more details. Have a web page with more details: preprints, project reports, source code, videos, etc. Is it possible and furthermore constructive to make LaTeX automatically format specific words in specific way? Let's say, for example, I'd like to have every occurrence of the word "hello" in a bold cyan monospaced font, whereas the rest of the document remains normal, sans serif and black.   How can I achieve that without having to do a search&amp;replace to place "hello" into a custom command every time before I compile the .tex file? How do you make the camera in blender cycles render everything so it looks like pixels? I want everything in my scene to look pixelly like this image I found: What is the path of an MTP device in Windows? I am running Windows XP and have a phone mounted through USB by MTP.  In Explorer, its address is:  My Computer\SAMSUNG-SGH-I547\Phone\DCIM   I would like to execute a command line command such as  explorer "My Computer\SAMSUNG-SGH-I547\Phone\DCIM\Camera"   so as to quickly go to the Camera folder. (Right-clicking the device or anything in it does NOT provide a "Create Shortcut" dialog.)  But "My Computer" is not a valid PATH in Windows. Unlike USB flash drives, it seems the phone mounted by MTP does not get a letter drive name.  So, what is the path of an MTP device in Windows?  If the answer is there is no command line path, is there workaround? None of the physics sensors in my game work, I'm trying to create an AI man that chases after a player, but for whatever reason, he just won't sense the player, I've tried copying and pasting the property name just to be sure, but that's not it. Powershell Try Catch Continue