Prove that $x^2+x+2$ is maximal ideal in $F_3[x]$ ring.  I think of using the theorem that $B$ is maximal ideal in $A$ if $A/B$ is field. Particularly in this case we need to prove that $F_3[x]/(x^2+x+2)$ is field. Could you please help?
Prove that $(x^2 + x+2)$ ideal is a maximal ideal in $F_3[x]$ ring.  I know the theorem about factor rings and fields     B ideal from A ring is maximal iff A/B factor-ring is a field   How can I use the theorem here?
A certain element which makes functionals positive Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there exist an element $x\in X$ such that $f_n(x) \in (0,\infty)$ for all $n$? Best way to obtain clarifying information when reputation doesn't allow comments? sed fails with "unknown option to `s'" error Is this correct?     That is a real cool answer.   I learned that that was incorrect, since "real" is an adjective which can describe a noun, e.g. "real answer" but it is not an adverb which can describe an adjective, "real cool". Instead you would have to say:     That is a really cool answer.   Since "really" is an adverb. Can I use Bluray as a boot cd? I'm trying to update an existing Subscriber in the All Subscriber Table through the API; i've set up the AppId in the App Center and have connected and created Subscribers, but i'm still having issues updating.  What call do I need to be making to do this? These are the ones i've been trying. Is the series $\sum_{1}^{\infty}\frac{1}{p_{j}}$ where $p_{j}$ is the $j$th prime convergent? Does the series $\sum_{n=1 }^{\infty}1/p_{j} $ of  reciprocal primes converge?  Experimentally, it seems convergent. Find executable files recursively I have got the directory called Test and a few directories inside it. Both Test and the directories inside it have executable files. I'd like to print them with ls. I'd use this command.  ls -l `find Test/ -perm /u=x,g=x,o=x -type f`   Is this a good/right/quick command or not?  My solution is:  find Test/ -executable -type f -exec ls -l {} \;   and got the same result as warl0ck and pradeepchhetri offered. I would like to run  something &gt; file   on a remote system through ssh, but if I run  ssh host something &gt; file   the redirection is executed locally as ssh etc &gt; file  I've tried it with ' or '' or dd or with a pipe | instead, but I can't get it to work. How can this be done? How to add a vertical line to a plot TikZ rectangular node with different rounded corners Are non-citizens of the Capitol permitted to permanently immigrate? Reading  got me thinking about permitted citizenship.  Are citizens of other districts allowed to permanently move to, or become citizens of the Capitol?   My gut says, "No." The citizens of the other districts are being punished, so it makes little sense to allow them a true, legal escape from their punishments. Even victors are only given new homes in their own districts, keeping them there. Which preposition is correct in the phrase "proficient in/at/with English"? How do I remove 'Places' entries from the Nautilus sidebar? Creating a copy of primary database and attaching the copy database in the same sql instance I just got into a project in which we have only one SQL Server 2012 copy. We have 12 databases attached to the primary instance of SQL Server 2012.  As I don't have SQL Server 2012 or supporting OS, I have to depend on the primary instance of SQL Server 2012.  I am not sure whether it is good to create copy/clone of all 12 databases and attach it to same instance and work on the copy for data manipulation and query testing purpose.  Based on Kin's query  SELECT    SERVERPROPERTY ('MachineName') AS PhysicalMachineName,    SERVERPROPERTY ('ServerName') AS SQLServerName,    SERVERPROPERTY ('Edition') AS ServerEdition,    CAST(SUBSTRING(@@Version,charindex('SQL',@@version, 1),15) AS VARCHAR(255)) +  ' + '         + CAST (SERVERPROPERTY ('productlevel')AS VARCHAR (50)) + ' + (Build'        + CAST (SERVERPROPERTY ('ProductVersion') AS VARCHAR (50)) +')' AS ProductVersion,     RIGHT(@@version, (Len(@@Version)-charindex('Windows',@@version, 1))+1) AS [O.S.],  SERVERPROPERTY ('Collation') AS Collation   Following is the output. I ignored PhysicalMachineName and SQLServerName.  ServerEdition : Business Intelligence Edition (64-bit) ProductVersion : SQL Server 2012 + RTM + (Build11.0.2100.60) O.S.           : Windows NT 6.1 &lt;X64&gt; (Build 7601: Service Pack 1)  Collation      : SQL_Latin1_General_CP1_CI_AS The cow in the field problem (intersecting circular areas) What length of rope should be used to tie a cow to an exterior fence post of a circular field so that the cow can only graze half of the grass within that field?  updated: To be clear: the cow should be tied to a post on the exterior of the field, not a post at the center of the field. I'm arriving from New Delhi, India (Indian passport) with JAL at Narita on 11th July at 07:30 and departing for San Francisco with JAL at 00:05 on 12th July from Haneda. Do I need a transit visa as I'll be switching airports and spending a few hours at JAL City Haneda Airport Hotel?  Previously I had connecting flights from Narita to San Francisco and obtained shore passes to relax at Narita Nikko Hotel. Now I'm travelling with my elderly Mom and do not want complications at Narita immigration or later in the day while going from Narita Airport to Haneda airport. Please advise. Keeping chunks loaded in minecraft I have built an iron farm somewhere far away from my actual spawn chunks in my single player world. In hindsight, this was a dumb idea, but I can't exactly change it anymore. I'd like to keep the thing running all the time, so I'm looking for a way to keep the chunks loaded.  I'm running minecraft 1.12.1, is it possible at the moment? I have looked at some chunk loader mods, but chicken chunk loader hasn't been updated yet, and the others simply didn't work.  I have tried moving the spawn chunks, but that hasn't done anything either.  Edit: I have worked this out with IronAnvil, and it ended up being a render setting in optifine which didn't load them correctly somehow. It has been working fine after I reset my optifine settings to default. I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation  $$D_{n,0}=\left[\frac{n!}{e}\right]$$  where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation that I wrote comes from solving the following recursion, but I don't understand how exactly the author calculated this recursion.  $$\begin {align*}  D_{n+2,0} &amp; =(n+1)(D_{n+1,0}+D_{n,0}) \\ D_{0,0} &amp; = 1 \\ D_{1,0} &amp; = 0  \end {align*} $$ In the movie Harry Potter and the Goblet of Fire, there's a scene where the students are preparing for the Yule Ball.   Professor McGonagall is instructing the boys and girls, while Filch is choosing a vinyl record to place in an enormous turntable phonograph with a horn as big as Hagrid.   This has me wondering if the wizarding world, in general, plays vinyl records. Or, if this was just a Muggle artifact that Hogwarts kept as a novelty.   Are there other instances of witches or wizards using or owning phonographs or record players in either the books or movies? If so, what are they? (Examples help to illustrate if this was a one-off or a trend)