Consider the polynomial ring $R=F[x,y]$, where $F$ is a field. Let $p(x)\in F[x]$ be an irreducible polynomial.  What are all the maximal ideals of $R$ that contain the principal ideal $p(x)R$?  My attempt:  By correspondence theorem for rings, this is equivalent to finding all maximal ideals of $R/p(x)R$. $p(x)R$ is a prime ideal of $R$, so $R/p(x)R$ is an integral domain. I am kind of stuck here.  This is a continuation, but not an exact duplicate of my previous question ().  Thanks for any help.    Update: I have an attempted solution using $R/p(x)R\cong K[y]$, where $K=F[x]/(p(x))$.  Then let $\pi:R\to K[y]$ be the quotient map. Then since the maximal ideals in $K[y]$ are $g(y)K[y]$, by correspondence theorem, the maximal ideals of $R$ containing $p(x)R$ are $$\pi^{-1}(g(y)K[y])$$ where $g(y)\in K[y]$ is an irreducible polynomial.  I am a little dissatisfied with my answer (mainly because of the $\pi^{-1}$), is there any more explicit representation of the maximal ideals $\pi^{-1}(g(y)K[y])$?
Let $K=F[x]/(p(x))$, where $F$ is a field, $p(x)\in F[x]$ is an irreducible polynomial.  What are all the maximal ideals in $K[y]$?  Attempt: I have an attempt, but not entirely sure it is correct.  Since $(p(x))$ is a maximal ideal in $F[x]$, this means $K$ is actually a field.  Thus $K[y]$ is a PID.  So maximal ideals in $K[y]$ are $(g(y))$ for irreducible polynomials $g(y)\in K[y]$?
A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$ Why did the Jedi want to bring balance to the Force? During the Prequel trilogy, the Jedi wanted to bring balance to the Force, in accordance with a prophecy.  There seemed to be a general agreement that this was a 'good thing', but a disagreement with the assertion that Anakin was the one to do it.  Why did the Jedi Council believe it was good to bring balance to the Force?  The Republic was mighty (in theory), the Council was established as THE source of Force users, the Sith were thought extinct, and the number of Dark Jedi was apparently very, very low.  It seemed, for all intents and purposes, as if the Force was strongly imbalanced in favor of the Light side...so why did the Council want to 'balance' it? Get current URL with jQuery? I am using jQuery. How do I get the path of the current URL and assign it to a variable?  Example URL:  http://localhost/menuname.de?foo=bar&amp;amp;number=0 I am trying to assign an image texture to an Image Texture node in 2.80. Very simple script - I'm just starting to learn how this stuff works.  This is what I have. It creates an image and and IT node, but it doesn't assign the one to other. Can anybody help? I've searched around, but can't find it in another question.  import bpy  #create blank image bpy.ops.image.new(name="TestImg", width=1024, height=1024)  #add image texture to object, to hold baked lighting mat = bpy.context.view_layer.objects.active.active_material image_node = mat.node_tree.nodes.new('ShaderNodeTexImage')  #assign image to image texture node image_node.image = "TestImg"   Thanks in advance. Someone's car hit our metal bar gate, so one of the rods is bent as shown in the photo. What kind of a tool can I use to bend this back, so it's straight? Thank you! Best way to capture historical changes between levels in a dimensional hierarchy What is the cleanest way to capture changes between levels in a dimensional hierarchy?  I have the dimensional hierarchy [Area] > [Region] > [Location], where Area is the parent to Region and Region is the parent to Location. Location is lowest level of the hierarchy, and is associated with the fact table.  The business rules are: Locations are able to change the Region they are assigned to based on business need. Furthermore, Regions are able to change the Area they are assigned to.  We want to capture those changes so that when we do historical analysis, we can compare business metrics before and after a Location changed its Region, and a before and after a Region changed its Area.  What is the best way to do this?  The four ways that I have come up with all seem to have major drawbacks:   Using a bridge table between hierarchy levels requires that you use time bounds (effective start/end dates) to correctly identify which parent a child belongs to as you walk up the hierarchy. That makes queries prone to error, especially with hierarchies with many levels. Versioning a child record whenever its relationship to its parent changes (e.g., versioning Region A when it moves from Area A to Area B) and then cascading that change all the way down the hierarchy (e.g., creating new versions of Locations tied to Region A so that they are tied to the new Region A). Updating the child record's reference to the parent record, making you use time bounds to reassemble the correct history (prone to error). Always use time bounds when querying up the hierarchy to make sure Location A is associated with the correct Region (likewise, Region A is associated with the correct Area) for the moment of the fact being analyzed.   None of these seems clean to me.  Is there a better way?  Thanks a lot What is Hawking radiation and how does it cause a black hole to evaporate? My understanding is that  isn't really radiated from a black hole, but rather occurs when a particle anti-particle pair spontaneously pop into existence, and before they can annihilate each other, the antiparticle gets sucked into the black hole while the particle escapes. In this way it appears that matter has escaped the black hole because it has lost some mass and that amount of mass is now zipping away from it.  Is this correct? If so, wouldn't it be equally likely that the particle be trapped in the black hole and the antiparticle go zipping away, appearing as if the black hole is spontaneously growing and emitting antimatter?  How is it that this process can become unbalanced and cause a black hole to eventually emerge from its event horizon and evaporate into cosmic soup over eons? I'm trying to update my system using apt-get, but it's giving me this error  Err http://bd.archive.ubuntu.com raring Release.gpg          Could not connect to bd.archive.ubuntu.com:80 (116.193.170.18), connection timed out Err http://bd.archive.ubuntu.com raring-updates Release.gpg   Unable to connect to bd.archive.ubuntu.com:http: Err http://bd.archive.ubuntu.com raring-backports Release.gpg   Unable to connect to bd.archive.ubuntu.com:http: Reading package lists... Done W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring/Release.gpg  Could not connect to bd.archive.ubuntu.com:80 (116.193.170.18), connection timed out W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring-updates/Release.gpg  Unable to connect to bd.archive.ubuntu.com:http: W: Failed to fetch http://bd.archive.ubuntu.com/ubuntu/dists/raring-backports/Release.gpg  Unable to connect to bd.archive.ubuntu.com:http: W: Some index files failed to download. They have been ignored, or old ones used instead.   I am not a Ubuntu master but I can understand this is not a computer error. Any clues on how to fix it? Is there a documentclass that produces 'endless' pages? Does anybody know of a documentclass that produces a page that has a fixed width and a variable length, depending on the actual content that needs to fit on the sheet? I'm trying to get something like a kind of scroll if possible. How can I remove duplicate values from a multi-dimensional array in PHP?  Example array:  Array (     [0] =&gt; Array     (         [0] =&gt; abc         [1] =&gt; def     )      [1] =&gt; Array     (         [0] =&gt; ghi         [1] =&gt; jkl     )      [2] =&gt; Array     (         [0] =&gt; mno         [1] =&gt; pql     )      [3] =&gt; Array     (         [0] =&gt; abc         [1] =&gt; def     )      [4] =&gt; Array     (         [0] =&gt; ghi         [1] =&gt; jkl     )      [5] =&gt; Array     (         [0] =&gt; mno         [1] =&gt; pql     )  ) When is $\sin(x)$ rational? What package should I use to typeset units? I know there are a few packages related to typesetting units:   unitsdef units, which can be seen as a "base package" used by unitsdef SIunits siunitx, which is the successor to SIunits maybe others? like cjwunits and hepunits   What is the difference between them? Do they also work inside Math mode? And, more importantly, which one should I use?  Suggestion for people trying to answer this question: show some examples of how to typeset both simple and complex units. For instance, try typesetting these items:   100 Celsius 3 x 10^5 km/s gravitational constant is measured in N m^2/kg^2 10 kHz = 1 s^{-4} sin(x) meters   I think these examples cover most needs when writing scientific texts: simple measurements (value + simple unit), values written in scientific notation, simple fractional units, complex fractional and squared units, units without any value accompanying them, units with arbitrary exponents, and arbitrary math code instead of a simple value. I was recently watching Episode 3 and a few clone wars episodes, and I found myself thinking during the opening battle between the Republic and General Grievous' forces above Coruscant, on a planet so densely populated where potentially thousands of inhabitants can be killed from a drifting capital class ship with a decaying orbit during space battles what defenses are in place?  Is there any instances of some form of emergency services that tractor beam chunks of debris to safe landing places or repulsor towers that can slow or suspend it until it can be taken apart safely? Even disintegration lasers or emergency force field generators like the types deployed on Hoth as they must have the energy output capable of sustaining it. This IS the galactic capital so must have some advanced tech at its deposal. Even now in our own development humans have certain preventative measures for all types of threats and ways of tracking them, plus responses or fall back plans. How to label using just numbers from start of text field? Using QGIS 3 graphical modeler I created a model with a variety of input parameters (> 10). When starting the model, a nice user interface is generated. But the arrangement of the GUI elements is totally confused and apparently unpredictable during creating the model. It does neither seem to depend on the arrangement of input parameters in the graphical modeler nor on the chronological order in which the input parameters are added to the model.   Is there a way that let's me arrange the GUI elements in the order I desire? The  EU book The Rise and Fall of Darth Vader reveals that Vader personally gave the stormtroopers the order to kill Owen and Beru Lars, and that he knew who he was condemning to death before he gave the order:  “Lord Vader,” said one of the sandtroopers, adjusting a control on his helmet so that only Vader could hear his voice.  “The Jawas sold a protocol droid and an astromech to these moisture farmers, but both droids are gone.” Moisture farmers? Intrigued, Vader examined the holograms of the kneeling couple and said, “The farmers’ names?” “Owen and Beru Lars, sir,” the sandtrooper responded. “They say they don’t know where the droids are, but it looks like a landspeeder is missing from their garage.” Owen and Beru, Vader recalled... “Tell Mr. and Mrs. Lars that they seem to have trouble keeping protocol droids on their property.”1 Not certain if he had heard correctly, the sandtrooper said, “Sir?” “Then you may extend to them every courtesy that you showed the Jawas before you continue your search. Establish checkpoints to detain any droids entering Mos Espa or Mos Eisley spaceports. And one more thing.” “Yes, Sir?” “Do not stop transmitting until I break the connection.” “Understood,” said the sandtrooper. Vader watched the sandtroopers carry out his orders on their helpless victims. He found the sight of rising flames — even holograms of flames burning millions of light years away — to be most satisfying.  I had always assumed that the stormtroopers killed the Larses on their own initiative, or in accordance with a previously given order.  The version of events in this book is interesting, but it raises the question of why it took Vader so long to figure out who Luke was. In canon, who gave the order to kill Owen and Beru Lars?  If Vader was involved, did he recognize the couple before he gave the order?  1 A reference to the fact that Anakin took C-3PO from the Lars homestead the last time he was on Tatooine, without informing the Larses that he would do so. StringTokenizer? Convert the String to a char[] and iterate over that? Something else? Reason for strange "canyons" of dust on the cooling platform I took apart my cooling platform Something like this :  And I found something like this inside :       I wonder why around the hollows are the "canyon like" structures.  PS: I also drop this platform on the floor, maybe this could be a reason ?   EDIT :  Everybody seems (except Brandon Enright answer) to ignore the word "dust" in this question. Those structures are not in the plastic itself. It is dust ON the plastic.  Proof :     Completely wiped out with my finger : I need help identyfying this movie I once saw late night on TNT, late 90s or early 2000s.  I don't have very clear recollection about it. The movie was very strange. It all took place in a single room of a spaceship or an escape pod that had lost power. A woman and a man, or several men, were stuck in a ship or a spacepod that was somehow damaged or didn't have power.   The movie was entirely made of the dialogs between the woman and the man (or men). They would talk about what was happening outside, how to contact help, about their lives. During their conversation details about the world around them emerge. In this movie people have colonized the solar system but not moved outside it, and there is some sort of a war.  Other things I remember are that it was dark in that room. I think they used only a lightbulb or even direct starlight and sunlight (if they were far away from the sun), because there was no power. Also it was getting colder, becaus the ship's heating system was not functioning, and so it was losing its heat. The people would constantly complain about how cold it's getting. I think that the woman was the captain of the ship. She would sometimes go to the control cabin and try to do something to help their situation, either restore power or contact someone so they can be rescued.  What impressed me about it was how they managed to make it realistic. Because the actors would talk and act like it was all real, it made it somehow feel very realistic to me as well.  That's about it as far as I remember. Hope someone has watched this movie and can tell me what it is.  Update  I want to add a few more details about the movie. While I watched it in the late 90s/early 2000s the movie looked much older. It looked like a vintage B-Movie made in the style of the 70s or 80s. In particular the decor looked more like Alien 1 or Cosmos 1999.   I just watched the firefly episode and it can't be it. It has too much action, and the atmosphere is not dark enough. Legend inconsistent in QGIS map print layout After I did styling my vector data set to be classified by graduated circle and categorized by type, it appears beautifully consistent in the map canvas as shown below:   But when I create a print map layout and after adding the map, I add the legend and it come out ugly inconsistent like this:    So, I would like to ask for help to make the legend in the map print layout consistent as the legend shown in layer panel. I use the latest version QGIS 3.10.2 64bit.