I am trying to construct a field with 27 elements, and find a primitive element in that field. I considered the irreducible polynomial $f(x)=x^3+2x+1$ over $\mathbb{Z}_3[x]$. Then I considered  $$\mathbb{Z}_3[x]/\langle f\rangle.$$ This is a field with $3^{\deg f}=3^3=27$ elements. I know that the unique elements of this field are given by  $$\{a_0+a_1t+a_2t^2:a_i\in\mathbb{Z}_3\}$$ where $t=x+\langle f\rangle$. Now my question is, what is an efficient way (for beginners) to find a primitive element of this field? One could argue that it suffices to find an element $u\in\mathbb{Z}_3[x]/\langle f\rangle$ with $\text{ord}(u)\neq 1,2,13$, but finding such a $u$ is computationally tedious (at least to me). Any comments or advice appreciated.
I have a finite field $\mathbb{F}_{27}$. I need to find an element of order 13. I know that multiplicative group of this field is cyclic with order 26. So I want to find a generator $g$ of this cyclic group, then $g^2$ will have order 13. Also I know that all elements of this field can be represented by polynomials from $\mathbb{F}_3[t]$, where t is root of irreducible polynom $x^3-x-1$.  Probably we can find generator of $\mathbb{Z}_{26}$ and than build an isomorphism to $\mathbb{F}_{27}\setminus 0$.  Anyway I don't know were to start.  Thanks!
using $$ in a post causes strange number series to appear What's a word for someone who is constantly asking for same thing? Hi I'm looking for a word for someone who is constantly asking for something and keeps wanting updates about it. The closest words I can think of is pushy and, to a lesser extent, annoying. But clearly these words don't directly mean what I intend. I really get confused when to use say and when to use tell. Which is appropriate in the following:   What did he tell? / What did he say? What are you saying? / What are you telling? Disable auto-opening nautilus window after auto-mount Every time I plug a USB stick in, nautilus opens a new window with the contents of the drive. I would like to disable this auto-opening of the nautilus window, but I would like the actual auto-mount to keep working. Is this possible? When rolling percentages, do 1d100 and two d10s (percentiles) share the same probabilities? I know d100s are shunned because they take too much time to roll, and 1d10+1percentile is really fun, but do they share the same probabilities? Is it better to use a (digital) d100?  Also, do d100s have 0? You can't roll a 0 with percentiles, I think. End of javascript sidebar workarounds? (Salesforce Classic / Aloha) Reading through this doc:      We replaced the rich text editor in HTML Area home page components   with a new version that supports more markup but doesn’t allow HTML to   be manually entered.      In Summer ’15 we will start removing unsupported code from HTML Area   home page components. As a result, components that contain JavaScript,   CSS, iframes, or other unsupported markup might stop working properly.   To use JavaScript or other advanced HTML elements in your home page   component, we recommend that you use a Visualforce Area component   instead.   With Visualforce homepage component the best we could do is design fancy home page components, which has not been the big use of sidebar amongst developers. How do we do the fancy hack we do today using html components since visualforce is hosted in a different domain will the sidebar hacks even work after summer 15?     You can include JavaScript in your Visualforce page, but because the   component is rendered in an iframe on the home page layout, the   JavaScript can’t interact with the page that contains the component.   Finally why this move :(? Evaluation Symbol: Consistent Implementation Column and row padding in tables Plural/singular verb agreement with units Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n&gt;2$? Why is inductance (L) proportional with turns-square (N²)? We start from the   $$ \mathbf{\nabla} \times \mathbf{B} = \mu \mathbf{J} + \overbrace{\mu \epsilon \dfrac{\partial \mathbf{E}}{\partial t}}^0. $$  We take surface integration of both sides, for the surface (\$s\$) inside the mean path (\$c\$) of the core.  $$ \int_s \left( \mathbf{\nabla} \times \mathbf{B} \right) \cdot d\mathbf{s} = \mu \int_s \mathbf{J} \cdot d\mathbf{s} $$  We use the  to rewrite the left hand side; where \$c\$ is in the same direction with the magnetic flux \$\Phi\$.  $$ \oint_c \mathbf{B} \cdot d \mathbf{\ell} = \mu N I $$  (The integral on the left hand side results \$NI\$, because there are \$N\$ different wires on the winding.)  The magnetic field density inside these kind of cores are considered to be uniform. So, we can write  $$ B \ell_c \overset\sim= \mu NI \implies B = \dfrac{\mu NI}{\ell_c}; $$  where \$\ell_c\$ is the mean path lenght of the core.  We can find magnetic flux from the magnetic flux density we have found by using the cross sectional area of the core \$A_c\$.  $$ \Phi = BA_c = \dfrac{\mu NIA_c}{\ell_c} $$  By definition, inductance is the amount of magnetic flux generated per applied current, that is  $$ L \overset\triangle= \dfrac{\Phi}{I}. $$  So, we find inductance of the system as  $$ \boxed{ L = \dfrac{\Phi}{I} = \dfrac{\dfrac{\mu NIA_c}{\ell_c}}{I} = \dfrac{\mu NA_c}{\ell_c} }. $$  But, all other sources () give inductance of an inductor like this as  $$ L = \dfrac{\mu N^2A_c}{\ell_c}. $$  What is the mistake I did in my derivation? Please explain in detail. I just spilt coffee on my laptop, what should I do? The Punjab is a rich state.   Is it correct to use the before Punjab? How to set the chapter style in report class? I want to set chapters to look like 'bigger' sections.  Now in the report class I have a style like:     Chapter 1      Introduction      1.1. Section   What I want to achieve is:        Introduction         1.1. Section   I don't want to change the document class. I have a web service in C# and would like to have a nested inner class, that abstracts away the session collection, something like this:     public class Service : System.Web.Services.WebService {     [WebMethod]     public string Foo(string ticket)     {         SessionPool.getSession(ticket);     }      private class SessionPool      {         public static Session getSession(string ticket)         {             // this is what i want to do, but I can't access Context             return (Session)Context.Session[ticket];         }     } }    Is it possible to access the HTTP context of the WebService class via a nested class? If not, is there way I can store the reference to it? Integral of type $\displaystyle \int\frac{1}{\sqrt[4]{x^4+1}}dx$ How do I make ditto marks? In France, students are taught not to repeat the same text twice when there is little difference between a line and the one below. It's a quick way to take notes.  I reproduced this effect by tinkering:    The code is as follows:  \documentclass[aspectratio=1610]{beamer}  \usepackage[T1]{fontenc} \usepackage[french]{babel}  \usepackage{amsmath, amsfonts, amssymb} \usefonttheme[onlymath]{serif}  \begin{document} \begin{frame} Lorsque le \og \textit{carreau carré} \fg{}   mesure: \begin{itemize} \item $\rm 1\,cm$ de côté, l'unité d'aire est appelée le \og \textrm{centimètre carré} \fg{} et est noté $\rm cm^2$; \item $\rm 1\,dm$ \hspace{1mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}''\hspace{8mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{décimètre carré} \fg{} \hspace{2mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1.5mm} $\rm dm^2$; \item $\rm 1\,m$ \hspace{3mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}'' \hspace{7mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{mètre carré} \fg{}  \hspace{9mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1mm} $\rm m^2$; \item etc. \end{itemize} \end{frame} \end{document}   How can I get the same effect more simply than by tinkering like I just did? Any nice, stable ways to keep a window 'Always on top' on the Mac? I am looking for a stable way to keep any window 'always on top' on OSX.  , an app that does that requires SIMBL/easySIMBL, which it seems is not very stable.  Are there any better solutions? It's actually surprising that there is no native way to implement this. Is every dense subspace of a separable space separable? Can "what kind" be plural? Is it correct to say, "What kind of patents are being issued in these sophisticated times?" It seems like it should be What kind of patents is being issued..., but that sounds wrong.  What kind of patent is being issued... is correct and sounds right, but loses a little bit of the original meaning.  Is there something magic about what kind that allows it to be plural, or do people just misuse it a lot?