Consider the polynomial $P=X^5-X-1\in\Bbb{F}_3[X]$. I want to show that $P$ is irreducible. We can easily check it has no roots, so the only way it could not be irreducible is by being a product of two polynomials of degree $2$ and $3$ respectively. So I determine all the irreducible polynomials of degree $2$ in $\Bbb{F}_3[X]$. These are $$X^2+1;{~~} X^2+X-1 {~~}\text{and}{~~}X^2-X-1.$$ Finally I check using the Euclidean division algorithm that none of those polynomials divides $P$, which concludes the proof.  My question is this:  is there a more efficient way to do this? I had to first determine all the polynomials of a given degree ($2$) and then apply Euclidean algorithm to each of them. That is quite some work, and we're still considering polynomials of relatively small degree, and fields of small cardinal. I can't imagine applying the same reasoning to determine whether $Q=X^9-X^2-1\in\Bbb{F}_{17}[X]$ is irreducible.  So, is there a more efficient method to determine irreducibility, at least in the case of polynomials of small degree over small fields?
Using the affine linearized polynomial $f(x)=x^p-x-b$ is irreducible over $\mathbb{F}_q$ if and only if $\operatorname{Tr}_{\mathbb{F}_q/\mathbb{F}_p} \neq 0$. But here $\mathbb{F}_3$ is not a field extension of $\mathbb{F}_5$.  Also we have that: Let $a \in \mathbb{F}_q$ and let $p$ be the characteristic of $\mathbb{F}_q$. Then the trinomial $x^P - x - a$ is irreducible in $\mathbb{F}_q[x]$ if and only if it has no root in $\mathbb{F}_q$.
How to detect if a variable is an array Find all integer solutions to $x^2+y^2+z^2=2xyz$ Determining a matrix from its characteristic polynomial Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like $$p(\lambda) = (-1)^n\lambda^n + (-1)^{n-1}(\text{tr } A)\lambda^{n-1} +  \cdots + \det A$$ Now, how can we obtain the matrix $A$ knowing the characteristic polynomial? Under Windows, most applications and application data are stored in a special directory known as C:\Program Files (and occasionally C:\Program Files (x86)). What is the Ubuntu/Linux equivalent to this path? Is there even one? Adding script conditionally I'm an Indian. I'm planning to book a journey from Mumbai to Johannesburg, South Africa and back (conference visa). I looked at refundable tickets, while they are priced higher, they give you half your money back, for instance if the to and fro flight costs INR 40,000 they give you INR 20,000 . But this is if you are cancelling for some reason.   What if you have to cancel because your visa application or the transit visa application got rejected. As far as I understand the procedure is :-  a. Buy airline tickets   b. Apply for Visa   c. Wait between 1-6 weeks to get update about your visa application, for both visa and transit visa (although transit visa should be shorter I guess.)  d. If visa is granted then move on to planning other things about the trip.   Now in case, if the visa application fails (for whatever reason ) what recourse does the traveler have ?  Are there any airlines who are better understanding than others about this situation ?  In non-refundable tickets, I guess that all is lost or is there some way to save money there too.   Lastly, I read travel insurance can help in this case. How does travel insurance help in the above scenario ?  Edit :- One of the pre-conditions of getting the visa is   Proof of a Valid Return or onward ticket or purchase of ticket. This is from   and Leftmargin problem at the bibliography I have some trouble with the bibliography alignment. It's similar to , but that solution doesn't work for me.   Here is MWE:  \documentclass{article}  \let\oldthebibliography=\thebibliography \let\endoldthebibliography=\endthebibliography \renewenvironment{thebibliography}[1]{%   \begin{oldthebibliography}{#1}%   \setlength{\parskip}{0ex}%   \setlength{\itemsep}{0ex}%   \setlength{\itemindent}{3ex}   \setlength{\leftmargin}{0pt}% }% { \end{oldthebibliography}% }  \begin{document}  Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text.    Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text.   \begin{thebibliography}{0} \bibitem{1} Baider A. Noncompact Riemannian manifolds with discrete spectra // J.Diff.Geom. -- 1979 -- V. 14 -- p. 41--57. \bibitem{2} Berger M., Gauduchon P., Mazet E. Le spectre d'une vari\'et\'e Rieman\-nienne -- Berlin-New York: Springer-Verlag, 1971. -- 251p. -- (Lecture Notes in Math.; V.~194) \end{thebibliography} \end{document}   So I need the reference labels to be aligned with the parindent, and the rest part of the each bibliography entry should have general text width. I thought it's enough to set \leftmargin=0pt, but it didn't work. I have the following result:   And what is actually more weird - it seems any value of \leftmargin doesn't have any effect to the bibliography display. The solution from the link above works fine if we don't have biblabel, but it's not my case.  Could you help me to move lines with the red marks to the green one? How can I get rid of this black hole in my minecraft world? I have interesting bug that I have experienced before but has gone away on its own. This one is not going away though.  I have a giant stripe or chunks missing from my desert biome, I would say probably 7 chunks long.  I'm guessing the chunks are corrupted or something, what was hoping to figure is whether it can repaired or reverted.  Here is a picture of the problem. Character Limit in HTML How do you impose a character limit on a text input in HTML? I have a virtual instance of redhat running that I can ssh to from any host on the corporate network.  My ssh session was dropped while running some scripts that may or not have caused me to disconnect.  The who command locally still lists the dropped connection. Reconnecting with ssh starts a whole new session.  Is there a way to connect to the original session?  If not can I kill it?  I have root and physical access.  I was not using screen in the original session. How does one actually show from associativity that one can drop parentheses? I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?  $$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$$  Do you just say that the grouping of the parentheses now corresponds to just multiplying straight through? Thanks. In blender cycles render engine, how do I put a movie clip on a plane so it displays the video? I want the video to be on the plane if there is a way. Compatible version of "Adobe Shockwave Player?" Is there a Ubuntu compatible version of the "Adobe Shockwave Player?" I'm running Firefox and when I try to bring up one of my School's websites it says missing plugin. When I click on "Install missing plug-in" it says none found, so when I click on manually install plug-in it takes me to Adobe Shockwave Player installation, but there is no ubuntu version to install??? On the Firefox plug-in finder service it says "Unknown Plugin (application/x-director) Is there an easier or more 'correct' way to open gedit as root? I have ubuntu 12.04 running in a virtual machine on virtual box.  Its existence is mainly as a 'toy' to play about with Ubuntu, so for this reason I wanted the GUI (is that what 'Dash' is in ubuntu 12.04?  Anyway...)  One thing I am currently using this machine for is a LAMP server.     I use 'gedit' to edit my files, but because I haven't correctly set permissions yet I just open it with sudo in terminal.  Just now, to try to simplify this process I created a shell script on the desktop with a very simple job - sudo gedit  I can't help thinking there must be an even more obvious way to open gedit as root from the GUI than this.  So is there?  Edit:  I just 'improved' my script by replacing sudo with gksudo.  Now I don't have to have a terminal window open while gedit is open.   So I guess this is quite convenient for now :)   It would be good, for the sake of knowledge, to know if this is anything close to how the more seasoned ubuntu users would do this type of thing. Software for text search in files I am looking for a software that will search text in files from a folder similar to .  Is there something similar? What is the definition of textbook or "raw" RSA?  What are some of the properties of textbook RSA?  How does it differ from other schemes based on RSA? How do I convert two lists into a dictionary? Imagine that you have the following list. keys = ['name', 'age', 'food'] values = ['Monty', 42, 'spam']  What is the simplest way to produce the following dictionary? a_dict = {'name': 'Monty', 'age': 42, 'food': 'spam'} Custom DataTable using Vanilla Datables in LWC I'm trying to build a custom datable because I want the functionality to sort, search and paginate and also add action buttons in each row. Initially, my VF page was using  but we have now decided to rebuild that page in LWC.  I found a similar  but it hasn't really worked for me.  I then found  (Javascript) and  (ES2018) I have tried to implement Vanilla Datatables but I keep running into issues such as:      TypeError: Cannot read property 'length' of undefined   The above error is thrown because the library is unable to find the body of the table.   My point is that I feel this library won't work because it's using document, window and so on.   I was wondering if there is a way I could convert this library to something that is more suited to LWC basically have it "export" the modules. Since I'm not a JS expert I'm unable to put the pieces together.   Just for some context, I'll post my code here:  &lt;template&gt;     &lt;table class="sobject-table slds-table slds-table_cell-buffer slds-table_bordered"&gt;         &lt;thead&gt;             &lt;tr class="slds-line-height_reset"&gt;                 &lt;template for:each={fieldSetData} for:item="field"&gt;                     &lt;th key={field.apiName} title={field.fieldLabel} class="" data-scope="col"&gt;                         &lt;div class="slds-truncate" title={field.fieldLabel}&gt;{field.fieldLabel}&lt;/div&gt;                     &lt;/th&gt;                 &lt;/template&gt;             &lt;/tr&gt;         &lt;/thead&gt;         &lt;tbody&gt;             &lt;template for:each={records} for:item="row"&gt;                 &lt;tr key={row} class="slds-hint-parent"&gt;                     &lt;template for:each={row} for:item="data"&gt;                         &lt;td key={data.id} data-label="Name" data-scope="row"&gt;{data.value}&lt;/td&gt;                     &lt;/template&gt;                 &lt;/tr&gt;             &lt;/template&gt;         &lt;/tbody&gt;     &lt;/table&gt; &lt;/template&gt;   import {LightningElement, api, wire} from 'lwc'; import {processRecords} from 'c/recordGenerator'; import getFieldSetData from '@salesforce/apex/FieldSetData.getFieldSetData'; import getSobjectRecords from '@salesforce/apex/DeomoCTRL.getRecords'; import VanillaDatatable from '@salesforce/resourceUrl/VanillaDatatable'; import {loadScript, loadStyle} from 'lightning/platformResourceLoader';  export default class ManageSelectedIndicators extends LightningElement {     @api recordId;     records;     fieldSetData;     @wire(         getFieldSetData,         {objectName: 'SOBJECTName', fieldSet: 'FIELSETName'}     ) wiredFieldSetData({error, data}) {         console.log('wired called in Child')         if (data) {             this.fieldSetData = data;             this.getData();         }     }       constructor() {         super();         console.log('Constructor called in Child');     }      connectedCallback() {         console.log('Connected callback');     }      renderedCallback() {         console.log('Rendered callback');         Promise.all([              loadScript(this, VanillaDatatable + '/VanillaDataTables/src/vanilla-dataTables.js')          ]).then(() =&gt; {              const table = new DataTable(                  document.getElementsByClassName('.sobject-table'),                  //Also tried this.template.querySelector('.sobject-table'),                  {searchable: true, sortable : true}              );          })          .catch(error =&gt; {             console.log(error);          });     }      getData() {         getSobjectRecords ({             sobjectId: this.recordId         })         .then(response =&gt; {             this.records= processRecords(response, this.fieldSetData);         })         .catch((error) =&gt; {             console.log(error);         });     }  }   Edit: I have also tried lwc:dom="manual" that didn't seem to work either.  Is there any other way I could achieve these functionalities? Or do I have to write custom code to achieve this? What is the subject of this sentence I've seen this sentence in someone's facebook status but it sounds unnatural to me.     I'm the Rumplestiltskin who spins straw into gold   I thought it should be spin because the subject is I, but I'm not sure. Which one is the subject in this sentence, I or Rumplestiltskin and why? How do I install LibreOffice 4? I want to download and install .  How can I do this?